Workspace Analysis

Problem

Given a robot and its joint limits, find the set of points in R³ that the TCP can reach. This set is called the work envelope, $Y$.

The shape of the work envelope in $R^3$ varies from robot to robot, so we describe the same reachable set using joint coordinates instead of cartesian coordinates.

$$Q=\{q\in R^n:q_{min}\le Cq\le q_{max}\}$$

• $Q$ is the joint-space work envelope
• $q$ is the joint variable vector, with n joints
• $q_{min}$ and $q_{max}$ are vectors in $R^m$ holding the lower and upper limits of how much the joints can rotate or translate
• $C$ is the $m\times n$ joint coupling matrix, and $m$ is the number of constraints

The limits of $q_{min}$ and $q_{max}$ of the joints apply to a linear combination of joint variables, not only to single joints

$p(q)$ maps a joint vector to the tool tip position in $R^3$. The work envelope $Y$ is the locus (set) of all points $p\in R^3$ that are reachable by the tool tip. It can be expressed in terms of $Q$:

$$Y=\{p(q)\in R^3:q\in Q\}$$

[! important] In simple terms, we are just trying to run forward kinematics once for every joint vector $q$ allowed by $Q$, and collect all the tool tip positions. That collection is our $Y$. In practice, we put joint limits into the forward kinematics expressions and work out the bounds, which is how we get $r_{max}$ and $r_{min}$ is later parts.

$p(q)$ is the fourth column of the arm matrix $T_{base}^{tool}(q)$ or equivalently the first three components of the tool-configuration vector $w(q)$.