Forward kinematics

Definition

Represent a rigid motion $x \mapsto R x + p$ of $\R^3$, with $R \in SO(3)$ and $p \in \R^3$, by the $4 \times 4$ matrix $\begin{pmatrix} R & p \\ 0 & 1 \end{pmatrix}$; these matrices form the Euclidean group $SE(3)$. Let a serial chain consist of links $0, 1, \dots, n$, with link $0$ fixed. Link $i - 1$ and link $i$ are joined by a joint with a real variable $q_i$ ranging over a set $Q_i \subseteq \R$. For each $i$, let $T_i \colon Q_i \to SE(3)$ give the pose of a frame fixed in link $i$ relative to a frame fixed in link $i - 1$. The forward kinematics map of the chain is the following map. $$f \colon Q_1 \times \cdots \times Q_n \to SE(3), \qquad f(q_1, \dots, q_n) = T_1(q_1)\, T_2(q_2) \cdots T_n(q_n)$$ Its value is the pose of the frame of link $n$ relative to the frame of link $0$, computed by matrix multiplication.