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 |