# 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 $$[Ad]^*$$