Geometry and assumptions
A planar gripper pose has three coordinates: x, y, and orientation φ. We use ideal revolute joints with no joint limits or collision checks. These are illustrative arms in arbitrary units, not SO-101 models.
Two joints can usually choose a reachable position, with a finite set of resulting orientations. Three joints can generally choose a reachable full pose, but the solution need not be unique. Four joints usually leave one degree of freedom while holding that pose fixed. These counts assume the relevant Jacobian has full rank; singularities and workspace boundaries need special care.
Link lengths: 2 joints = (0.62, 0.44); 3 joints = (0.44, 0.36, 0.26); 4 joints = (0.32, 0.28, 0.24, 0.22). The shaded region is the position workspace, not the set of positions reachable at every approach angle.
φ = q₁ + q₂ + … + qₙ
Null-space motion: J(q) q̇ = 0
For four joints, the slider varies the first joint over one connected feasible interval and solves the remaining three joints analytically. It shows one solution family, not every branch. The same local redundancy idea applies to a seven-joint arm controlling a six-dimensional gripper pose.