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Performance Evaluation of a Class of Gravity-Compensated Gear-Spring Planar Articulated Manipulators

  • Nguyen Vu Linh
  • Chin-Hsing KuoEmail author
Conference paper
  • 665 Downloads
Part of the Mechanisms and Machine Science book series (Mechan. Machine Science, volume 78)

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 analysis 

1 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

The 1-, 2-, and 3-DOF gear-spring articulated manipulators in Ref. [15] are taken as examples for illustrating the performance of gravity compensation, as shown in Fig. 1(a), (b), and (c), respectively. In these drawings, mi, si, li, and θi represent the equivalent mass, center of mass, length, and rotation angle of link i (i = 1, 2,…, n) of the manipulator, respectively; d0i, dsi, ψi, ngi, and ki represent the spring compression at the initial position, slider displacement, assemblage angle between HiAi and the rotating link, gear ratio, and spring stiffness of GSM i, respectively; φ0i and φci represent the initial and instantaneous deflection angles of the connecting rod AiBi with respect to the rotating link OiBi of GSM i, respectively; r2ai = HiAi and r3i = AiBi.
Fig. 1.

The studied gear-spring articulated manipulators with (a) one DOF, (b) two DOFs, and (c) three DOFs.

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

For each manipulator as mentioned above, the evaluation is performed for both design approaches. The criterion used for the evaluation is defined as follows:
$$ \delta_{ti} = \left[ {1 - \frac{{\text{max}\left( {\left| {T_{m/si} \left( {\theta_{i} } \right)} \right|} \right)}}{{\text{max}\left( {\left| {T_{mi} \left( {\theta_{i} } \right)} \right|} \right)}}} \right] \times 100\% $$
(1)
where Tmi, Tm/si, and δti represent the motor torques of the manipulator without, with gravity compensation, and the peak-to-peak torque reduction at the joint Oi (i = 1, 2, 3) of the manipulator, respectively.

3.1 The 1-DOF Articulated Manipulator

Using the Ideal Balancing Approximation.

Assume that the design parameters of the 1-DOF articulated manipulator are given in Table 1. From these data, the influence of the ratio r31/r2a1 to the peak-to-peak torque reduction δt1 can be illustrated in Fig. 2(a). As may be seen, increasing the ratio leads to improve the efficiency of torque reduction and δt1 can reach almost 100% when the ratio approaches infinity, as the ideal case. In order to obtain δt1 ≥ 90%, the ratio is expected to be greater than 12 or r31 ≥ 0.48 m. However, this desirable parameter can make the connecting rod lengthy as compared to the radii of gears.
Table 1.

Parameters of the 1-DOF gear-spring articulated manipulator.

ψ1 (°)

n g1

r11 (m)

r21 (m)

r2a1 (m)

m1 (kg)

s1 (m)

90

2

0.02

0.04

0.04

2.44

0.135

Fig. 2.

Torque reduction of the 1-DOF gear-spring articulated manipulator versus (a) ratio r31/r2a1 and (b) coefficient ηs1.

On the other hand, consider a case when the length of the connecting rod is prescribed, for example, r31 = 0.18 m, whereas the coefficient ηs1 is variable in the interval [0.8, 1.6]. Then, the peak-to-peak torque reduction δt1 can be computed versus the coefficient and the results are shown in Fig. 2(b). It can be seen that high torque reduction (δt1 ≥ 90%) can be acquired whenever ηs1 is selected from 1.2 to 1.3, even the ratio here r31/r2a1 = 4.5. The highest value δt1 = 94.1% is obtained when ηs1 = 1.23. Compared to the previous case, it is clear that selecting an appropriate coefficient ηs1 = [1.2, 1.3] is more convenient than using a long-length connecting rod (r31/r2a1 ≥ 12) since both approaches could produce δt1 ≥ 90%. From the obtained parameters when δt1 is maximum, some other weights m1 were simulated and verified on MSC Adams; indeed, the results of δt1 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: r2a1 = 0.04 m, d01 = 0, ks1 =  = [4000, 8000] N/m, r31/r2a1 = [1, 5], ψ1 = [0°, 360°], ng1 = [1, 3], and θ1 = [0°, 180°]. The optimization was then performed and terminated within 35 iterations at δt1 = 96.1%, whereas the optimal design parameters were also obtained as: ks1 = 6325 N/m, r31/r2a1 = 3.69, ψ1 = 91.42°, and ng1 = 2.01.

Based on the above results, the motor torques of the manipulator by using the ideal balancing approximation Tm/s1 (appro.) and the realistic optimization Tm/s1 (opti.) can be illustrated in Fig. 3. Note that only the case with the highest value δt1 shown in Fig. 2(b) is taken in the representation for the approximation approach. The results show that Tm/s1 (appro.) and Tm/s1 (opti.) are much smaller than the motor torque without gravity compensation Tm1 at almost rotation angle θ1. Besides, δt1 can be slightly improved from 94.1% (by the approximation) to 96.1% when the optimal design parameters are used. For reducing the computational effort, it is therefore preferable adopting the approximation approach for designing the 1-DOF articulated manipulator.
Fig. 3.

Motor torques of the 1-DOF gear-spring articulated manipulator.

3.2 The 2-DOF Articulated Manipulator

Similar to the previous section, a numerical example of the 2-DOF articulated manipulator integrated with two GSMs for gravity compensation is illustrated in this section. The performance evaluation of the manipulator via two described design approaches is also presented. Assume that the design parameters of the manipulator and GSMs used in the approximation are given in Tables 2 and 3, respectively. The manipulator is supposed to perform tasks on two predefined square trajectories D1D2D3D4 and E1E2E3E4 with the side length of 0.4 m and constant speed of 0.16 m/s (the total time is 10 s), as shown in Fig. 4(a) and (b), respectively. Note that trajectory 1 is with link 1 of the manipulator in the vertical direction, while trajectory 2 with an inclined angle 30° at the initial position. On the other hand, the optimal design parameters of the manipulator for the two trajectories were also derived, as listed in Table 3.
Table 2.

Parameters of the 2-DOF gear-spring articulated manipulator.

m1 (kg)

m2 (kg)

s1 (m)

s2 (m)

l1 (m)

l2 (m)

5

3

0.4

0.3

1

0.8

Table 3.

Parameters of the GSMs used in the 2-DOF gear-spring articulated manipulator (The parameters listed in Tables 3 and 5 are obtained from Ref. [15].).

Approach

GSM i

ψi (°)

n gi

r1i (m)

r2ai (m)

r3i (m)

ki (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 × 104

4.486 × 103

Optimization

Trajectory 1 - D1D2D3D4

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 × 104

6.199 × 103

Trajectory 2 - E1E2E3E4

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 × 104

6.059 × 103

Fig. 4.

Trajectory examples of the 2-DOF gear-spring articulated manipulator.

The motor torques of the 2-DOF articulated manipulator on the two prescribed trajectories can be illustrated in Fig. 5(a) and (b), respectively. As can be seen, the motor torques at the joints (θ1, θ2) are significantly reduced when using either the approximation or optimization. It is interesting to observe that the peak-to-peak torque reduction at the two joints δti (i = 1, 2) can be improved when the optimal design parameters are used. On the first trajectory, δt1 is slightly improved from 95% to 97.7% at the first joint while an increase of δt2 by 21% can be seen at the second joint. On the other hand, the improvement is little on the second trajectory, being 6.6% and 4.2% at the first and second joints, respectively.
Fig. 5.

Motor torques of the 2-DOF gear-spring articulated manipulator versus (a) trajectory D1D2D3D4 and (b) trajectory E1E2E3E4.

3.3 The 3-DOF Articulated Manipulator

Similarly, the performance evaluation of the 3-DOF articulated manipulator integrated with three GSMs is presented in this section. Assume that the design parameters of the manipulator and GSMs used in the manipulator are listed in Tables 4 and 5, respectively. The manipulator is assumed to perform tasks on two predefined square trajectories F1F2F3F4 and G1G2G3G4 with the side length of 0.4 m and constant speed of 0.16 m/s, as shown in Fig. 6(a) and (b), respectively. Besides, the manipulator is also assumed to maintain link 3 in the horizontal direction during the first trajectory while an orientation 45° from the vertical direction during the second trajectory.
Table 4.

Parameters of the 3-DOF gear-spring articulated manipulator.

m1 (kg)

m2 (kg)

m3 (kg)

s1 (m)

s2 (m)

s3 (m)

l1 (m)

l2 (m)

l3 (m)

5

3

1.5

0.4

0.3

0.2

1

0.8

0.5

Table 5.

Parameters of the GSMs used in the 3-DOF gear-spring articulated manipulator.

Approach

GSM i

ψi (°)

n gi

r1i (m)

r2ai (m)

r3i (m)

ki (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 × 104

1.196 × 104

2.659 × 103

Optimization

Trajectory 1 - F1F2F3F4

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 × 104

9.068 × 103

3.213 × 103

Trajectory 2 - G1G2G3G4

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 × 104

9.625 × 103

2.263 × 103

Fig. 6.

Trajectory examples of the 3-DOF gear-spring articulated manipulator.

The motor torques of the 3-DOF articulated manipulator on the two prescribed trajectories are illustrated in Fig. 7(a) and (b), respectively. As compared to the case without gravity compensation, the motor torques at the first two joints (θ1, θ2) are significantly reduced by using the approximation, and that decreased almost half at the third joint (θ3). The peak-to-peak torque reduction δti (i = 1, 2, 3) can be improved when the optimal design parameters are used. On the first trajectory, the torque reduction by using the optimization is slightly improved by 7% and 2.8% at the first and second joints, respectively. The similar results at the first two joints of the manipulator can also be observed on the second trajectory. In contrast, there is a significant improvement observed at the third joint, being 46.6% and 57.4% on the first and second trajectories, respectively.
Fig. 7.

Motor torques of the 3-DOF gear-spring articulated manipulator versus (a) trajectory F1F2F3F4 and (b) trajectory G1G2G3G4.

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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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.National Taiwan University of Science and TechnologyTaipeiTaiwan
  2. 2.University of WollongongWollongongAustralia

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