FORCE-DISTRIBUTION ANALYSIS FOR REDUNDANT CABLE-DRIVEN PARALLEL ROBOTS UNDER HYBRID JOINT-SPACE INPUT
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摘要: 绳驱并联机器人是由绳索代替刚性杆件的一类特殊机器人, 其中绳索具有只能承受拉力而不能承受压力的特点, 冗余绳驱并联机器人的绳力分配问题是一个难点. 在关节混合空间控制中, 将冗余的绳索组合采用绳力控制, 而其余绳索进行绳长控制. 因为不同的绳索组合可能导致不同的控制效果, 本研究旨在解决关节混合空间控制条件下, 力控绳索组合的选择问题. 以二冗余绳驱并联机器人为例, 通过向量空间基变换方法, 实现了冗余绳驱系统绳力在拉力索张力空间的表达. 基于拉力索张力空间, 计算了绳力控制绳索组合的对称最大误差带, 用于找到合适的绳索组合用于力控. 使用多体动力学仿真手段, 对关节混合空间的控制效果和对称最大误差带解析解计算方法的正确性进行了模拟验证. 在同时考虑绳长和绳力控制误差的条件下, 发现当选择不合适的绳索组合时, 绳力误差会被显著放大, 说明了本文针对绳力分布特性分析的意义. 本文提出的对称最大误差带概念同时也为关节混合空间控制策略下的绳力控制器设计提供指导.Abstract: Cable-driven parallel robots (CDPRs) represent a class of particular parallel robots whose rigid links are replaced by cables, where cable can only generate pull force and cannot be compressed. The force distribution in cables is one of the core problems for redundant CDPRs. The hybrid joint-space control strategy, where the chosen redundant cables are force-controlled, whereas the remaining ones are length-controlled in the joint space, is the main type of control strategy discussed in this paper. Because different cable combinations may lead to different control effects. This study provides the selection criteria for the target force-controlled cable combination in the hybrid-input control strategy. The cable tensions in the space with tension vectors as basis for two redundancies for cable-driven parallel robots were expressed based on the equivalent transformation method of vector space basis. The acceptable cable force errors limit proposed in this paper (CFEL) were defined and calculated based on the cable tensions in the space with tension vectors to find appropriate cable combinations for the force control. To validate the analysis of the force-distribution characteristics, a hybrid-input control trajectory planning strategy was developed using multibody dynamics simulations, based on a suspended cable configuration with layouts including two redundancies, while considering the interference of cable length and cable forces. In addition, a fix-pose simulation case via hybrid-input control strategy was performed to validate accuracy of the proposed calculation method for the CFEL. Finally, we found that cable combinations play an essential role for force control as the force control errors may be significantly magnified in cable combinations with high force-distribution sensitivity characteristics. The simulation results illustrate the significance of the analysis in this paper. What’s more, the concept of CFEL proposed in this paper provides guidance for the design of cable force controllers under the control strategies of hybrid joint-space input.
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图 2 不等式组的平行线几何模型解释
Figure 2. Explanation of the geometric model of parallel lines for the set of inequalities
图 3
${{ {\boldsymbol{T}}}_1} - {{ {\boldsymbol{T}}}_2}$ 空间内的绳力可行域, 黑色边界代表绳力可行域的边界, 正方形S1S2S3S4代表选择绳索组合1–2作为力控时的最大绳力误差区域Figure 3. The TFR in T1–T2 space, black line represents the boundaries of the TFR, and square S1S2S3S4 represents the maximum cable force error region for cable combination 1–2
图 4 对称最大误差带的计算
Figure 4. Geometric interpretation for calculating the cable force error limit for two redundancies
图 5 模拟案例中的两冗余度绳驱并联机器人, 数字代表绳索编号
Figure 5. Layout of the cable suspended robots. Numbers represent cable identifications
图 7 采用绳索组合7–8力控时的对称最大误差带计算
Figure 7. Calculation of the CFEL when choosing the cable combination 7–8
7 采用绳索组合7–8力控时的对称最大误差带计算(续)
7. Calculation of the CFEL when choosing the cable combination 7–8 (continued)
图 9 末端平台的(a)位置, (b)速度和(c)加速度
Figure 9. Desired (a) position, (b) velocity and (c) acceleration profiles of the end-effector
图 11 (a)绳长控制随机误差(仅取一条绳索为例)和(b)两冗余条件下绳力控制随机误差
Figure 11. The control errors: (a) represents the length error (choose one cable as an example) and (b) represents the force errors for the chosen cable combination
图 12 绳力分布曲线: (a)理想绳力分布, (b)采用绳索组合7–8力控时没有绳长控制误差的绳力分布, (c)采用绳索组合7–8力控时绳长误差和绳力误差共同作用的绳力分布, (d)采用绳索组合3–7力控时没有绳长控制误差的绳力分布, (e)采用绳索组合3–7力控时绳长误差和绳力误差共同作用的绳力分布
Figure 12. Force distribution: (a) ideal force distribution, (b) force distribution without cable length error when cable combination 7–8 is force-controlled, (c) force distribution with cable length error when cable combination 7–8 is force-controlled, (d) force distribution without cable length error when cable combination 3–7 is force-controlled and (e) force distribution with cable length error when cable combination 3–7 is force-controlled
图 13 控制误差对整体绳力分布的影响: (a)采用绳索组合7–8进行力控和(b)采用绳索组合3–7(对照组)进行力控
Figure 13. Influence of the overall cable distribution of tensions: (a) and (b) represent cable combination 7–8 and cable combination 3–7 (force-controlled), respectively
图 14 绳索组合4–8的力控灵敏系数分析: (a)表示绳索4的力控灵敏系数和(b)表示绳索8的力控灵敏系数
Figure 14. Calculation of FCS for cable combination 4-8: (a) and (b) represent the FCS of cable 4 and 8, respectively
表 1 出绳点和结绳点的位置信息
Table 1. Position parameters of the cable robots
Cable The positions of fixed points in the inertial frame O (X, Y, Z) The positions of distal points in the inertial frame O’ (x, y, z) point X/m Y/m Z/m point x/m y/m z/m 1 A1 1.992 −0.174 2.000 B1 0.1075 −0.1687 0 2 A2 1.992 0.174 2.000 B2 0.1075 0.1687 0 3 A3 −0.845 1.813 2.000 B3 0.0923 −0.1774 0 4 A4 −1.147 1.638 2.000 B4 −0.1998 0.0087 0 5 A5 −1.147 −1.638 2.000 B5 −0.1998 −0.0087 0 6 A6 −0.845 −1.813 2.000 B6 0.0923 −0.1774 0 7 A7 −2.000 0 2.000 B7 −0.1998 −0.0087 0 8 A8 1.000 1.730 2.000 B8 0.1075 0.1687 0 
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表 2 在[0 0 0]处绳索组合及其对应的FDS和CFEL值
Table 2. The FDS and CFEL values at the position [0 0 0]
No. i j FDS CFEL Tmin = 1 N CFEL Tmin = 2 N CFEL Tmin = 3 N No. i j FDS CFEL Tmin = 1 N CFEL Tmin = 2 N CFEL Tmin = 3 N 1 1 2 12.49 0.66 0.55 0.45 15 3 5 2.32 3.31 2.72 2.14 2 1 3 5.96 1.44 1.21 0.97 16 3 6 6.56 1.31 1.10 0.88 3 1 4 5.86 1.31 1.08 0.85 17 3 7 2.26 3.40 2.80 2.20 4 1 5 10.24 0.66 0.55 0.43 18 3 8 8.48 1.01 0.84 0.68 5 1 6 139.84 0.04 0.036 0.029 19 4 5 6.51 1.18 0.93 0.76 6 1 7 6.49 1.17 0.97 0.76 20 4 6 6.91 1.11 0.92 0.72 7 1 8 7.21 1.19 1.00 0.80 21 4 7 10.24 0.75 0.62 0.49 8 2 3 6.19 1.39 1.16 0.93 22 4 8 2.37 3.62 3.03 2.43 9 2 4 2.47 2.74 2.25 1.77 23 5 6 12.84 0.53 0.44 0.35 10 2 5 2.92 2.06 1.70 1.33 24 5 7 9.32 0.83 0.68 0.53 11 2 6 12.79 0.65 0.54 0.44 25 5 8 2.23 3.74 3.12 2.51 12 2 7 2.37 2.95 2.43 1.91 26 6 7 7.91 0.97 0.80 0.63 13 2 8 10.23 0.84 0.70 0.56 27 6 8 7.89 1.09 0.91 0.73 14 3 4 2.95 2.62 2.16 1.70 28 7 8 0.70 5.96 4.91 3.86
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