Abstract
This paper deals with a control scheme for underwater vehicle-manipulator systems with the dynamics of thrusters in the presence of uncertainties in system parameters. We have developed two controllers that overcome thruster nonlinearities, which cause an uncontrollable system: one is a regressor-based adaptive controller and the other is a robust controller. However, the structure of the adaptive controller is very complex due to the feedforward terms including the regressors of dynamic system models, and the error feedback gains of the robust controller with a good control performance are excessively high due to the lack of feedforward terms. In this paper we develop an adaptive controller that uses radial basis function networks instead of the feedforward terms. The replacement leads to a moderately high gain controller whose structure is simpler than that of the regressor-based adaptive controller.
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Abbreviations
- \( N_{\text{v}} \) :
-
Number of dimensions for motion of vehicle
- \( N_{\text{e}} \) :
-
Number of dimensions for motion of manipulator end-effector
- \( N_{n} \) :
-
Number of dimensions for motion of vehicle and manipulator end-effector (i.e., \( N_{\text{v}} + N_{\text{e}} \))
- \( N_{\text{a}} \) :
-
Number of dimensions for vehicle’s orientation and manipulator’s joint angles
- \( N_{\text{m}} \) :
-
Number of links (we will assume in Sect. 2 that \( N_{\text{m}} = N_{\text{e}} \))
- \( x\left( t \right) \) :
-
Signal composed of vehicle’s and manipulator end-effector’s positions and orientations (\( \in R^{{N_{n} }} \))
- \( \phi (t) \) :
-
Signal composed of vehicle’s orientation and manipulator’s joint angles \( \left( { \in R^{{N_{\text{a}} }} } \right) \)
- \( u(t) \) :
-
Signal composed of vehicle’s translational velocity and \( \dot{\phi }(t) \) \( \left( { \in R^{{N_{n} }} } \right) \)
- \( \tau_{\text{m}} (t) \) :
-
Joint torques of manipulator \( \left( { \in R^{{N_{\text{m}} }} } \right) \)
- \( J(\phi ) \) :
-
Jacobian matrix in the equation \( \dot{x}(t) = J(\phi )u(t) \) \( \left( { \in R^{{N_{n} \times N_{n} }} } \right) \)
- \( \bar{R}(\phi ) \) :
-
Transformation matrix from thrust forces to force and torque concerning inertial coordinate system \( \left( { \in R^{{N_{\text{v}} \times N_{\text{v}} }} } \right) \)
- \( \bar{M}(\phi ) \) :
-
Inertia matrix \( \left( { \in R^{{N_{n} \times N_{n} }} } \right) \)
- \( \bar{f}( \cdot ) \) :
-
Signal composed of centrifugal, Coriolis, gravitational and buoyant forces, and fluid drag \( \left( { \in R^{{N_{n} }} } \right) \)
- \( v(t) \) :
-
Shaft velocities of thruster’s propellers \( \left( { = [v_{1} (t), \ldots ,v_{{N_{\text{v}} }} (t)]^{\text{T}} \in R^{{N_{\text{v}} }} } \right) \)
- \( \tau_{\text{b}} (t) \) :
-
Shaft torques of thruster’s propellers \( \left( { \in R^{{N_{\text{v}} }} } \right) \)
- \( A,\;B,\;\bar{K} \) :
-
Diagonal matrices composed of thruster’s system parameters \( \left( { \in R^{{N_{\text{v}} \times N_{\text{v}} }} } \right) \)
- \( \text{sgn} ( \cdot ) \) :
-
Signum function
- \( \exp ( \cdot ) \) :
-
Exponential function (e.g., \( \exp (x) = e^{x} \))
- \( {\text{diag}}\{ \cdot \} \) :
-
Diagonal matrix (e.g., \( {\text{diag}}\{ a_{1} , \ldots ,a_{n} \} \) where \( a_{i} \) are diagonal elements)
- \( {\text{tr}}\{ \cdot \} \) :
-
Trace of matrix
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Taira, Y., Oya, M. & Sagara, S. Adaptive control of underwater vehicle-manipulator systems using radial basis function networks. Artif Life Robotics 17, 123–129 (2012). https://doi.org/10.1007/s10015-012-0023-7
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DOI: https://doi.org/10.1007/s10015-012-0023-7