Notation

Conventions

example description reserved for
\(A\) capital symbols Matrices
\(a\) lowercase symbols Scalar quantities
\(\underline{a}\) underlined symbols Vector quantities, or a column matrix of stacked vector quantities
\(\dot{v}\) overhead dot First time derivative
\(\ddot{v}\) overhead double dot Second time derivative
\(\begin{bmatrix} \end{bmatrix}\) square brackets \(m \times n\) matrices where \(m\) and \(n\) are typically greater than 1
\(\begin{Bmatrix} \end{Bmatrix}\) curly brackets or braces \(m \times 1\) column matrix
\(\mathbf{\underline{v}}\) bold face/underlined \(6 \times 1\) screw quantities, such as velocities and accelerations
\(\mathbf{\underline{f}}^*\) bold face, underlined, asterisk \(6 \times 1\) coscrew quantities, such as forces.

Symbols

symbol name in program description
\(\ddot{q}\) ddq second time derivative of generalized coordinates
\(\dot{q}\) dq first time derivative of generalized coordinates
PC PC parent child list
\(\lambda_p\) lambda_p lagrange multipliers
\(Q\) Q working forces applied in the joints
\(\gamma^p\) gamma_p quadratic velocity terms
\(M^{\ell}\) M_l generalized mass matrix
\(\lambda^d\) lambda_d preconditioned lagrange multipliers
\(\gamma^d\) gamma_d preconditioned quadratic velocity terms
\(f^{\ell}\) f_l generalized forces
\(J\) J Jacobian matrix, \(J(q)\), a mapping of body velocities to generalized velocities
\(\underline{\mathbf{p}}^*\)   momentum
\(\underline{\mathbf{h}}\) h influence coefficient matrix
\(\underline{\mathbf{v}}\) v velocity screw \(\left(\underline{\mathbf{v}} = \begin{Bmatrix}\underline{v}\\\underline{\omega}\end{Bmatrix}\right)\)
\(\underline{\mathbf{g}}^*\) g externally applied forces, such as from gravity

Transformations

symbol operation description
\(R\)   \(R \in \mathbb{R}^{3x3}\), an SO(3) rotation matrix
\(R^{-1}\) \(R^T\)  
\(\underline{r}\)   \(r \in \mathbb{R}^{3}\), a translation vector
\(\underline{r}^{-1}\) \(- \underline{r}\)  
\(\tilde{r}\) \(\begin{bmatrix}0&-r_z&r_y&\\r_z&0&-r_x\\-r_y&r_x&0\end{bmatrix}\) \(\tilde{r} \in \mathbb{R}^{3x3}\), a skew-symmetric matrix
\(\Phi\) \(\begin{bmatrix}R&\underline{r}\\0 & 1\end{bmatrix}\) \(\Phi(R,r) \in SE(3)\), a general rigid body displacement consisting of rotation (\(R\)) and translation (\(r\)).
\(\Phi^{-1}\) \(\begin{bmatrix}R^T&-R^T\underline{r}\\0&1\end{bmatrix}\)  
\([Ad]\) \(\begin{bmatrix}R & \tilde{r}R\\0&R\end{bmatrix}\) adjoint operator for transforming screws
\([Ad]^{-1}\) \(\begin{bmatrix}R^T&-\tilde{r}R^T\\0&R^T\end{bmatrix}\) inverse of adjoint operator for transforming screws
\([Ad]^*\) \(\begin{bmatrix}R & 0\\\tilde{r}R&R\end{bmatrix}\) coadjoint operator for transforming coscrews
\([ad]\) \(\begin{bmatrix}\tilde{\omega}&\tilde{v}\\0&\tilde{\omega}\end{bmatrix}\) time derivative of \([Ad]\)
\([ad]^*\) \(\begin{bmatrix}\tilde{\omega}&0\\\tilde{v}&\tilde{\omega}\end{bmatrix}\) time derivative of \([Ad]^*\)