Online Learning Control of Uncertain Nonlinear Systems Using Gaussian Processes and Its Application
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摘要: 针对一类不确定非线性系统, 提出了一种基于高斯过程的在线学习控制方法. 该方法首先通过障碍函数间接设定系统状态的运行区域. 其次, 在该区域内在线采集量测数据, 利用高斯过程回归对系统中未知非线性动态进行学习. 然后通过Lyapunov稳定理论, 证明了所提在线学习控制算法可保证闭环系统所有信号的有界性. 与基于径向基神经网络(Radial basis function neural networks, RBFNNs) 的自适应控制方案相比, 所提控制算法无需精确给出系统状态的运行区域及预先分配径向基函数中心值. 最后, 通过数值仿真与Franka Emika Panda 协作机械臂关节控制实验, 验证了控制算法的有效性与先进性.Abstract: This paper presents a Gaussian process-based online learning control (GP-OLC) approach for a class of uncertain nonlinear systems. Initially, a barrier function is introduced to indirectly define the operating domain of the system states. Subsequently, a Gaussian process regression is employed to learn the unknown system dynamics utilizing measurement data collected online within this domain. Through Lyapunov stability theory, we prove that the proposed GP-OLC algorithm ensures the boundedness of all signals in the closed-loop system. Compared to adaptive control schemes using radial basis function neural networks (RBFNNs), the developed controller does not necessitate precise specification of the system states' operating domain nor preallocation of radial basis function (RBF) center values. Finally, the effectiveness and superiority of the proposed approach are validated through numerical simulations and joint control experiments on a Franka Emika Panda collaborative robot.
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Key words:
- Nonlinear systems /
- uncertain systems /
- Gaussian processes /
- online learning control /
- manipulators
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图 7 由Franka Emika Panda机械臂本体和控制箱组成的实验平台
Fig. 7 The experimental platform consisted of the Franka Emika Panda robot body和control box
图 8 位置状态$q_1$, 参考轨迹$q_{d1}$和跟踪误差$e_{11}=q_1-q_{d1}$
Fig. 8 Position state $q_1$, desired trajectory $q_{d1}$ and tracking error $e_{11}=q_1-q_{d1}$
图 9 位置状态$q_2$, 参考轨迹$q_{d2}$和跟踪误差$e_{12}=q_2-q_{d2}$
Fig. 9 Position state $q_2$, desired trajectory $q_{d2}$ and tracking error $e_{12}=q_2-q_{d2}$
图 11 文中GP-OLC和PD控制作用下关节位置跟踪误差$e_{11}=q_1-q_{d1}$
Fig. 11 Position tracking error $e_{11}=q_1-q_{d1}$ under the proposed GP-OLC in this paper and PD control
图 12 文中GP-OLC和PD控制作用下关节位置跟踪误差$e_{12}=q_2-q_{d2}$
Fig. 12 Position tracking error $e_{12}=q_2-q_{d2}$ the proposed GP-OLC in this paper and PD control
表 1 在时间间隔$ [20,\;30] $上跟踪误差$ e_1 $和$ e_2 $的$ L_2 $范数
Table 1 $ L_2 $ norm of tracking errors $ e_1 $ and $ e_2 $ over time interval $ [20,\;30] $
GP-OLC GP-OLFLC PID RBFNNs-AC $ ||e_1||_{L_2} $ 4.46 5.41 6.76 97.22 $ ||e_2||_{L_2} $ 2.61 2.82 32.41 161.17 
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表 2 Franka Emika Panda机械臂的运动学参数
Table 2 Kinematic parameters of Franka Emika Panda robot
关节 $ j $ $ d_j $[m] $ a_j $[rad] $ b_j $[m] $ q_j $[rad] 关节1 0.333 0 0 $ q_1 $ 关节2 0 $ -\dfrac{\pi}{2} $ 0 $ q_2 $ 关节3 0.316 $ \dfrac{\pi}{2} $ 0 $ q_3 $ 关节4 0 $ \dfrac{\pi}{2} $ 0.0825 $ q_4 $ 关节5 0.384 $ -\dfrac{\pi}{2} $ − 0.0825 $ q_5 $ 关节6 0 $ \dfrac{\pi}{2} $ 0 $ q_6 $ 关节7 0 $ \dfrac{\pi}{2} $ 0.088 $ q_7 $
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