Continuum Robot Control Based on Varying Parameter Recursive Network and Recursive Least Square
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摘要: 连续体机器人通常由柔性材料制成, 能够承受大幅度形变, 在各领域具有广阔的应用前景. 然而, 其软体结构和非传统的驱动机制也带来了诸多非线性因素, 使得其状态和运动难以被精确建模. 因此, 为连续体机器人设计了一种无模型控制方案. 该方案一方面通过变参递归神经网络求解连续体机器人的逆运动学, 以实现高精度运动控制, 另一方面使用递归最小二乘法基于实时数据估计和更新机器人雅可比矩阵伪逆, 以避免机器人的解析建模. 最后, 通过仿真模拟和实物实验验证了所提出控制方案的可行性、精确性和鲁棒性, 并通过一系列对比实验突出了所提出方法的优势. 该方法率先研究基于递归最小二乘法的连续体机器人雅可比矩阵伪逆估计, 对未来的连续体机器人研究具有一定的启示作用.Abstract: Continuum robots are usually made of soft materials and can withstand significant deformation, making them promising for applications in various fields. However, their soft structures and non-traditional actuation mechanisms also bring many nonlinear factors, leading to difficulties in modeling their states and motions precisely. Therefore, this work designs a model-free control scheme for continuum robots. On the one hand, the scheme solves the inverse kinematics of continuum robots through a varying parameter recursive neural network to achieve high-precision motion control. On the other hand, it uses the recursive least square method to estimate and update the pseudo-inverse of Jacobian matrix based on real-time data, for the sake of avoiding analytical modeling of robots. Finally, simulations and physical experiments are performed to verify the feasibility, accuracy and robustness of the proposed scheme, and the merits of the proposed method are revealed through a series of comparative experiments. This work pioneers the study of Jacobian matrix pseudo-inverse estimation for continuum robots based on the recursive least square method, which could inspire the future research on continuum robots.
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图 3 仿真跟踪结果((a) 采用三段连续体机器人, 利用所提出的基于改进动态变参数增强零化神经网络的控制方案仿真得到的轨迹; (b) 跟踪误差变化$ {\boldsymbol{e}}=[e_x,\; e_y,\; e_z]^{\rm{T}} $; (c) 状态参数q的变化过程)
Fig. 3 Simulation tracking results ((a) Trajectory of the three-segment continuum robot using the proposed control scheme based on Adapted-DVPEZNN; (b) Tracking error $ {\boldsymbol{e}}=[e_x,\; e_y,\; e_z]^{\rm{T}} $; (c) The change of state parameter q)
表 1 不同ZNN模型设计公式及激活函数对比
Table 1 Comparison of design formulas and activation functions of different ZNN models
模型 设计公式 激活函数 Original-ZNN[21] $ -\lambda \Phi\left({\boldsymbol{e}}\right) $ $ \phi\left(e\right)=e $ VP-CDNN[24] $ -\lambda \exp(t) \Phi\left({\boldsymbol{e}}\right) $ $ \phi\left(e\right)=e $ FTC-ZNN[25] $ -\lambda \left(P({\boldsymbol{e}})+\displaystyle \int_0^t Q({\boldsymbol{e}}(\tau)){\rm{d}}\tau\right) $ $ \left\{ \begin{aligned} &p(e)=((k_1g^\frac{1}{2}(t,\; e)\left|e\right|^\frac{1}{2})+(k_2g(t,\; e)\left|e\right|)){\rm{sign}}(e)\\&q(e)=((k_3g(t,\; e)+(k_4g^2(t,\; e)\left|e\right|){\rm{sign}}(e) \end{aligned} \right. $ CVP-RNN[33] $ -\lambda (t)P({\boldsymbol{e}}(t))-\displaystyle \int_0^t \mu(\tau)Q({\boldsymbol{e}}(\tau)){\rm{d}}\tau $ $ \left\{ \begin{aligned} &p(e)=\frac{1}{\sigma}\exp\left(\left|e\right|^\sigma\right)\psi^{1-\sigma}(e)\\ &q(e)=\frac{1}{\sigma}\exp\left(2\left|e\right|^\sigma\right)\left(\psi^{1-\sigma}(e)+\frac{1-\sigma}{\sigma}\psi^{1-2\sigma}(e)\right) \end{aligned} \right. $ DVPEZNN[34] $ -\lambda\exp\left(\left(\beta^t+\beta\right)\Vert{\boldsymbol{e}}\Vert_2\right)P({\boldsymbol{e}}) $ $ p(e)= \begin{cases} \zeta_1\psi^{r_1}(e)+\zeta_2e\exp\left(\left|e\right|+1\right),\; \left|e\right|\leq1\\\zeta_1\psi^{r_2}(e)+\zeta_2e\exp\left(\left|e\right|+1\right),\; {\text{其他}} \end{cases} $ Adapted-DVPEZNN $ \begin{aligned} & -\lambda\big(\exp\left(\xi_1\beta^t\Vert{\boldsymbol{e}}\Vert_2\right)P({\boldsymbol{e}})+\\ &\displaystyle \int_0^t \exp\left(\xi_2 \tau +\xi_3\right)Q({\boldsymbol{e}}) {\rm{d}}\tau \big) \end{aligned} $ $ \left\{\begin{aligned} &p(e)= \begin{cases} \zeta_1\psi^{r_1}(e)+\zeta_2e\exp\left(\zeta_3\left|e\right|+1\right),\; \left|e\right|\leq1\\\zeta_1\psi^{r_2}(e)+\zeta_2e\exp\left(\zeta_3\left|e\right|+1\right),\; {\text{其他}} \end{cases}\\ &q(e)= \begin{cases} \zeta_4\psi^{r_1}(e),\; \left|e\right|\leq1\\\zeta_4\psi^{r_2}(e),\; {\text{其他}} \end{cases} \end{aligned} \right. $ 
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表 4 参数更新方法的定量分析
Table 4 Quantitative analysis of parameter update methods
方法 均方误差(mm) 控制能耗(s) 增量法 0.033 33.658 双模型法 0.058 1 356.239 递归最小二乘法 0.057 7.689
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表 5 收敛系数$ \lambda $定量分析
Table 5 Quantitative analysis of convergence factor $ \lambda $
收敛系数$ \lambda $ 均方误差(mm) 控制能耗(s) 1 0.150 7.872 3 0.122 7.786 5 0.110 7.830 10 0.093 7.607 30 0.068 7.426 50 0.057 7.382
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表 6 遗忘因子$ \gamma $定量分析
Table 6 Quantitative analysis of forgetting factor $ \gamma $
遗忘因子$ \gamma $ 均方误差(mm) 控制能耗(s) 0.1 5.762 $ \times 10^{-2} $ 7.454 0.3 5.757 $ \times 10^{-2} $ 7.508 0.5 5.759 $ \times 10^{-2} $ 7.408 0.7 5.751 $ \times 10^{-2} $ 7.368 0.9 5.711 $ \times 10^{-2} $ 7.382 1.0 5.939 $ \times 10^{-2} $ 7.779
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表 7 不同场景下的跟踪误差定量分析
Table 7 Quantitative analysis of tracking errors in different scenarios
场景设置 均方误差 (mm) 无负载无障碍物 0.854 1个负载无障碍物 0.986 2个负载无障碍物 1.208 无负载有障碍物 0.869
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