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Table 1 Description of key notations and abbreviations

From: Extended DMPs Framework for Position and Decoupled Quaternion Learning and Generalization

\(\{ \cdot \}\) Trajectory from one demonstration \(\{ \{ \cdot \} \}\) Multi-trajectories from multi-demonstrations
\({\varvec{p}}\) Position \({\varvec{q}}\) Quaternion
\(\theta\) Angle-quaternion \({\varvec{v}}\) Axis-quaternion
\({\varvec{T}}\) Time \({\varvec{s}}\) Phase variable
\({\varvec{\varPsi}}\) RBFs \(W\) Weights of RBFs
\(c_{i}\) Center of i-th RBFs \(h_{i}\) Width of i-th RBFs
\({\varvec{R}}_{O}^{{\hat{O}}}\) Rotation matrix from \(O\) to \(\hat{O}\) \({\varvec{\gamma}}\) Vectors from \({\varvec{v}}_{i}\) to \({\varvec{v}}_{{i{ + 1}}}\)
\(f(s)\) Nonlinear term \(\tau\) Temporal scaling factor
\({\text{d}}({\varvec{x}},{\varvec{y}})\) Geodesics between x and y M Number of points in a demonstration
K Number of demonstrations N Number of Gaussian distributions
\({\varvec{\xi}}^{I}\) Inputs of GMR \({\varvec{\xi}}^{O}\) Outputs of GMR
\(P( * )\) Probability distribution \(\uppi\) Probability of Gaussian distributions
\({\varvec{\mu}}\) Mean of Gaussian distributions \(\user2{\sum }\) Covariance of Gaussian distributions
\(e_{c}\) Absolute error of Cartesian skills \(\Delta e_{c}\) Relative error of Cartesian skills
\(e_{r}\) Absolute error of Riemannian skills \(\rho_{c}\) PCCc of Cartesian skills
\(\rho_{r}\) PCCr of Riemannian skills \(\sigma\) Standard deviation
DMPs Dynamic movement primitives GMM/R Gaussian mixture model/ regression
TS Transformation system EM Expectation-maximization
RBFs Radial basis functions LWR Locally weighted regression
PCCc Pearson’s correlation coefficient in Cartesian space PCCr Pearson’s correlation coefficient on 2D sphere manifold
R Reproduced curve G1−3 Generalized curve 1-3