Support vector machine networks for friction modeling

Support Vector Networks in Adaptive Friction Compensation

“…At this stage, we make the following assumption and remark: Assumption: The matrix for the support vector set is strictly positive definite. Remark 3: This assumption trivially holds for commonly used kernels including the spline kernel in the sequel [19].…”

“…This constrained optimal problem can be solved by the standard technique of Lagrangian multipliers. As a result, the vector of the coefficients is the solution of the following quadratic program (QP) problem [19]: (17) subject to (18) Remark 2: Compared with the standard SVR [3], our augmented formulation (17) uses different regularization parameters and the -insensitive levels. The pairs and correspond to the smoothing efforts and the error-tolerances imposed on in the low-and high-velocity regimes, respectively.…”

“…Motivated by the downward bend behavior of the Stribeck effect, the straightforward selection of is determined by (12) As in [19], is approximated underlying the training data set by (13) with being a kernel function defined over . This representation is motivated by seeking a solution of the following regularization problem: (14) in a reproducing kernel Hilbert space (RKHS) defined by kernel [23].…”

“…The basic idea is to construct the approximation of from observed data. The SVR-based method developed in [19] is suitable for this task. We briefly describe the SVR formulation in the context of the friction model structure (6).…”

“…However, these studies mainly focus on the adaptation and stability of adaptive compensators without special considerations on the construction of NNs. In [19], SVR parametrization has been developed for friction modeling. It has been shown that there is no need to seek complex sensing techniques to collect the training data and thus the SVR parametrization can be easily implemented for a servo motion system such as a haptic display system [20].…”