2.2. Workspaces

Workspace and Configuration Space

The configuration space $\mathcal{Q}$ of a robot is the set of all admissible joint vectors $\Theta$, accounting for joint limits and constraints. For a serial manipulator with $n$ independent joints, $\mathcal{Q}\subset {\mathbb{R}}^n$.
The workspace is the image of the forward kinematic map:
$W=\{g(\Theta )\in SE(3)\mid \Theta \in \mathcal{Q}\}.$
It consists of all end-effector poses that the robot can achieve.

Two important subsets are distinguished:

• Reachable workspace: all points in ${\mathbb{R}}^3$ that the end-effector can physically reach, regardless of orientation.
• Dexterous workspace: the subset of the reachable workspace where every orientation in $SO(3)$ is attainable.

For many manipulators with two main links of lengths ${\ell }_1$ and ${\ell }_2$, the reachable workspace of the wrist center forms a spherical shell with outer radius ${\ell }_1+{\ell }_2$ and inner radius $|{\ell }_1-{\ell }_2|$, ignoring joint limits and self-collisions. The dexterous workspace is generally smaller and depends on whether full orientation freedom is available at each point.

The dimension of the workspace is at most six (three translational and three rotational degrees of freedom), even if the robot has more than six joints. When $n>6$, the manipulator is redundant, and multiple configurations correspond to the same end-effector pose.

Intuition

The configuration space $\mathcal{Q}$ and workspace $W$ are related by the forward kinematic map $g:\mathcal{Q}\to SE(3)$. The key insight is that $\mathcal{Q}$ is always $n$-dimensional (one dimension per joint), while $W$ maxes out at 6 dimensions regardless of $n$. When $n>6$, the forward map is many-to-one, and the preimage of a single pose is a continuous manifold of dimension $n-6$—these are the self-motions exploited by redundant manipulators.

The boundary of the reachable workspace is where the Jacobian loses rank: at these configurations, the manipulator is at a singularity and cannot move outward. This connects workspace geometry to differential kinematics.

Properties

(Workspace Boundary) The boundary of the reachable workspace corresponds to singular configurations of the manipulator. A point on the workspace boundary can only be reached with the arm fully extended (or in some other singular pose).

(Redundancy and Self-Motion) For $n>6$, the null space of the Jacobian $\mathcal{N}(J)$ has dimension $n-6$ at generic configurations. Self-motions change the joint configuration without moving the end-effector—the arm “reconfigures” internally while the tool stays fixed.