Whole-Body Control: Redundant Arm Null-Space (2D)

Target

Target X 2.20

Target Y (height) 1.60

Drag the yellow target dot directly on the canvas, or use the sliders.

Redundancy Resolution

Null-space gain k 1.20

Live Readouts

End-effector error

0.000 m

Worst joint margin

–

Null-space use

0.00

Legend

Arm link Target (drag) Joint — safe margin Joint — near its limit

What this 2D companion changes

The 3D scene resolves the same kind of redundancy — a 2D task with more joints than it needs — by steering the elbow away from a movable obstacle. This companion keeps the identical mathematics (damped pseudo-inverse primary task, null-space-projected secondary task) but points the secondary objective at a completely different criterion: each of the 4 joints has its own mechanical limit range, and the secondary task climbs the gradient of

H(q) = −(1/2N) · Σᵢ ((qᵢ − midᵢ) / rangeᵢ)²

— maximized when every joint sits at the centre of its own travel. The resulting joint velocity is Δq = J⁺·e + (I − J⁺J)·k·∇H: the primary term keeps the hand locked on the target exactly as before, and the null-space term (analytic gradient, not finite-difference) continuously nudges every joint back toward mid-range using only the motion the reach task doesn't need.

Reset pose starts three of the four joints deliberately close to their limits, so with the toggle off you can watch Naive mode leave them there (or drive them further in) while it only cares about the end-effector.
Null-space gain k — how hard the secondary term pulls joints toward centre; too high and it can slow how quickly the arm settles after you move the target.
Joint bars panel below the arm shows each joint's live position inside its own [min, max] range — the bar turns red inside the danger band near either end.

Real redundant manipulators use exactly this criterion (Liegeois' joint-limit-avoidance gradient) stacked below the reach task, so an arm can keep working near full extension without ever slamming a joint into its hard stop.

4-joint planar arm

Drag the target dot

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Joint position within its own limit range

Whole-Body Control: Redundant Arm Null-Space Simulator (2D)