Obstacle avoidance of a seven-joint manipulator based on double interpolation trajectory control
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摘要: 基于七關節機械臂的解析解提出了一種新的基于雙插補的機械臂軌跡控制算法，該方法利用給定的機械臂腕部關節中心點的位置向量推導出該位置向量在軌跡規劃器TP中的插補運算方程，并依據腕部中心點位置向量得到第七關節自運動的旋角，將每個插補周期插補出的旋角值附加到已通過解析解計算出的逆解關節向量的第七關節角度值上，從而達到關節軌跡控制的目的. 此外還可以通過通用的梯度投影法計算出一組關節角度值作為參考來進行解析解選解，從而達到避免奇異點的影響的效果. 該方法基于改造后的七關節機械臂構型在LinuxCNC實時控制平臺和matlab仿真平臺上都進行了實驗，并以機械臂的末端精度作為衡量指標，驗證了該方法對機械臂關節軌跡控制的有效性，也體現了其相比于梯度投影法等常用算法末端精度大幅提升的誤差控制優越性.Abstract: In recent years, with advancements in industrial production requirements, the seven-joint manipulator—a representative of intelligent and automatic technology—has received extensive attention from scholars locally and abroad. One of the most fundamental technological aspects in the field of seven-joint manipulator research is the obstacle avoidance and trajectory control technology of the manipulator. In the field of obstacle avoidance of robotic arms, despite the fact that the traditional gradient projection method based on the obstacle avoidance index can achieve the purpose of obstacle avoidance, several challenges will continue to be encountered, including the decline of end tracking accuracy and the inability to accurately control the trajectory of joint linkage. Because of the aforementioned shortcomings, this paper proposes a novel manipulator trajectory control algorithm based on double interpolation, which is, in turn, based on the analytical solution of the seven-joint manipulator. This method utilizes the given position vector of the wrist joint center point of the manipulator to derive the interpolation operation equation of the position vector in the trajectory planner, thereby obtaining the rotation angle of the seventh joint self-motion according to the position vector of the wrist center point. The rotation value interpolated by each interpolation period is added to the seventh joint-angle value of the inverse solution joint vector calculated by the analytical solution. This leads to achieving the purpose of joint trajectory control. The double interpolation manipulator trajectory control algorithm can also calculate a group of joint-angle values using the general gradient projection method as a reference, conducting analytical solution selection to circumvent singular point effects. The new double interpolation manipulator trajectory control algorithm based on the modified seven-joint manipulator configuration has been tested on the LinuxCNC real-time control platform and the Matlab simulation platform, and the end accuracy of the manipulator is taken as the measurement index to verify both the effectiveness of the joint trajectory control of the manipulator based on this method and the superiority of the end accuracy control compared with the gradient projection method. After verification, the accuracy of the double interpolation trajectory control algorithm proposed in this paper is found to be at least 3.59 × 10?8 and 2.74 × 10?8 orders of magnitude higher than that of the gradient projection method and the singular robust inverse algorithm, respectively. In the case of a complex end motion trajectory, such as an arc trajectory, its accuracy may even be improved by 10?11 orders of magnitude. In summary, the trajectory control method based on double interpolation can be applied to the obstacle avoidance scenario without affecting the end trajectory because of its strong error control capability.
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Key words:
- double imputation /
- trajectory control /
- obstacle avoidance /
- seven-joint robotic arm /
- end precision
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圖 1 七關節機械臂仿真模型及關節結構圖像. (a) 仿真模型; (b) 簡化的關節結構圖
Figure 1. Simulation model and joint structure image of a seven-joint manipulator: (a) simulation model; (b) simplified joint structure diagram
圖 5 機械臂從初始位姿運動到目標位姿. （a）初始位姿;（b）目標位姿
Figure 5. Robotic arm from the initial pose to the target pose: (a) initial pose; (b) target pose
圖 6 分解成兩個自運動的示意圖及其對應的節點向量圖. （a）簡化機械臂分解成兩個自運動示意圖；（b）節點向量圖
Figure 6. Schematic of decomposition into two self-motions and their corresponding node vector diagrams: (a) schematic of the simplified robotic arm decomposition into two self-motions; (b) node vector diagram
圖 8 采用關節連桿軌跡控制算法前后的機械臂運動軌跡圖.（a）未用關節連桿軌跡控制算法的機械臂運動軌跡圖；（b）采用關節連桿軌跡控制算法的機械臂運動軌跡圖
Figure 8. Motion trajectories of the manipulator before and after using the joint-link trajectory control algorithm: (a) motion trajectory of the manipulator without the joint-link trajectory control algorithm; (b) motion trajectory of the manipulator with the joint-link trajectory control algorithm
圖 9 分別使用梯度投影法和關節連桿軌跡控制算法的機械臂末端誤差圖.（a）使用梯度投影法的機械臂末端誤差圖；（b）使用軌跡控制算法的機械臂末端誤差圖
Figure 9. Error diagrams of the end of the manipulator using the gradient projection method and the joint-link trajectory control algorithm: (a) error diagram of the end of the manipulator using the gradient projection method; (b) error diagram of manipulator end using the trajectory control algorithm
圖 11 基于雙插補的關節連桿軌跡控制法效果圖. （a）三維仿真模型姿態；（b）實際機械臂構型
Figure 11. Effect diagram of the joint-link trajectory control method based on double interpolation: (a) pose of the 3D simulation model; (b) configuration of the actual manipulator
表 1 機械臂D-H參數表
Table 1. D–H parameter table of the robotic arm
Connecting rod coordinate
system, iJoint angle, ${\theta _i}$ Link length, ${a_{i - 1}}$/mm Link torsion angle, ${\alpha _{i - 1}}$ Link offset, ${d_i}$/mm 1 π/2 0 0 0 2 ?π/2 0 ?π/2 0 3 0 360 0 0 4 0 90 ?π/2 376.5 5 0 0 π/2 0 6 0 0 ?π/2 192.5 7 0 0 ?π/2 52 
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表 2 三種算法誤差對比表
Table 2. Error comparison table of three algorithms
Track type Number of interpolation
pointsError Error reduction factor Gradient
projection methodSingular robustness inverse Trajectory control algorithm Compared to gradient projection method Compared to singular robustness inverse Straight trajectory 300 $ 3.774 \times {10^{ - 4}} $ $ 4.461 \times {10^{ - 4}} $ $6.878 \times {10^{ - 12}}$ $1.54 \times {10^{ - 8}}$ $1.82 \times {10^{ - 8}}$ 500 $ 5.316 \times {10^{ - 4}} $ $ 6.418 \times {10^{ - 4}} $ $1.143 \times {10^{ - 11}}$ $2.15 \times {10^{ - 8}}$ $1.78 \times {10^{ - 8}}$ 800 $6.37 \times {10^{ - 4}}$ $8.015 \times {10^{ - 4}}$ $1.825 \times {10^{ - 11}}$ $2.86 \times {10^{ - 8}}$ $2.28 \times {10^{ - 8}}$ 1000 $ 6.327 \times {10^{ - 4}} $ $ 8.289 \times {10^{ - 4}} $ $2.274 \times {10^{ - 11}}$ $3.59 \times {10^{ - 8}}$ $2.74 \times {10^{ - 8}}$ Arc trajectory 300 $ 2.084 \times {10^{ - 2}} $ $ 2.159 \times {10^{ - 2}} $ $ 2.274 \times {10^{ - 13}} $ $1.09 \times {10^{ - 11}}$ $1.05 \times {10^{ - 11}}$ 500 $ 1.25 \times {10^{ - 2}} $ $ 1.295 \times {10^{ - 2}} $ $ 1.705 \times {10^{ - 13}} $ $1.36 \times {10^{ - 11}}$ $1.32 \times {10^{ - 11}}$ 800 $ 7.81 \times {10^{ - 3}} $ $ 8.095 \times {10^{ - 3}} $ $ 1.137 \times {10^{ - 13}} $ $1.46 \times {10^{ - 11}}$ $1.40 \times {10^{ - 11}}$ 1000 $ 6.248 \times {10^{ - 3}} $ $ 6.476 \times {10^{ - 3}} $ $ 5.684 \times {10^{ - 14}} $ $9.10 \times {10^{ - 12}}$ $8.78 \times {10^{ - 12}}$
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