Reachable workspace

Reachability is a set, not one distance limit

The Modern Robotics definition separates a robot's task space from its workspace. Task space describes the mathematical space in which a configuration or end-effector pose lives. The reachable workspace is the subset the mechanism can actually attain under its allowed joint configurations.

For a simple planar arm, link lengths create familiar inner and outer boundaries. A real humanoid arm adds joint limits, coupled shoulders, torso geometry, and orientation requirements. A point may be reachable by the wrist while the required wrist orientation is not.

Constraints create several useful workspaces

“Reachable” should state which constraints are included. A geometric workspace may ignore collision and torque. A collision-free workspace excludes poses that intersect the robot or environment. A balanced whole-body workspace can be smaller again because the feet, centre of mass, and support contacts constrain the posture.

The term therefore does not mean that every point in the set is equally easy to reach. Near a kinematic singularity, a pose can be reachable while small task-space changes require large joint motion or produce poor numerical conditioning.

Inverse kinematics tests a target locally

Inverse kinematics searches for a configuration that realises a requested pose. A successful solve is evidence that the target lies in the workspace represented by that model and constraint set. A failed numerical solve is not always proof that the target is outside it, because the result can depend on the starting configuration, solver, tolerances, and local minima.

For a motion, every pose must be considered with its neighbours. Solving each frame independently can produce discontinuous joints even when every wrist target is individually reachable. A useful trajectory also needs speed, collision, and dynamic checks.

Base placement changes a humanoid's task-relative workspace

A mobile or free-standing humanoid can translate its whole body relative to the object. That does not change the arm's body-relative workspace, but it changes which world-frame targets fall inside it.

EmbodyHOI optimises the standing position together with arm joints while guiding a hand–object motion. The shorter Unitree G1 moves closer to the object, while the longer-armed H1-2 moves back. This illustrates why a data record should keep the robot model, object frame, base pose, joint constraints, IK residuals, and trajectory together.

Reachability remains a kinematic property. It does not establish that the robot can generate the required forces, maintain contact, balance, or execute the path on hardware.