Joint angles define a point in configuration space. The reachable workspace traces an envelope. Inverse kinematics becomes a visual search problem in parallel coordinates.
A robot arm with N revolute joints has N joint angles. Each configuration is a point in N-dimensional configuration space. Forward kinematics maps it to the end-effector position.
In parallel coordinates each configuration is a polyline across N joint-angle axes. Adding axes for the end-effector position extends the polyline to encode both input and output of the forward kinematics map.
For a planar arm with three links of lengths , , , the forward kinematics are:
Each choice of joint angles determines a unique end-effector position. The inverse problem — finding joint angles that reach a target — generally has multiple solutions.
Inverse kinematics in parallel coordinates is a search: sample many configurations as polylines across joint-angle and end-effector axes, then select the polylines that pass through the desired end-effector region.
Drag the target in the arm view, or adjust link lengths and tolerance.
Figure 1. A planar 3-link arm. Drag the target to set a desired end-effector position. All configurations that reach within tolerance of the target are shown as polylines on the three joint-angle axes. The manifold of inverse kinematics solutions is the family of polylines.
For targets deep in the reachable workspace, many configurations succeed and the polyline family is dense. As the target nears the workspace boundary, the family thins. Beyond it, nothing survives.
Sampling configurations uniformly at random and plotting each as a polyline across all five axes (three joint angles plus two end-effector coordinates) gives a dense picture of how joint space maps to workspace.
Brush on the x or y axes to select configurations that reach a region of workspace. The selected polylines show which joint angle combinations achieve that reach.
Figure 2. 500 random configurations on 5 parallel axes. Brush on the end-effector axes (x, y) to isolate configurations that reach a particular workspace region. The remaining polyline segments reveal the joint-angle structure.
Brushing the end-effector axes and reading the joint-angle axes is inverse kinematics — algebraic inversion becomes geometric selection.
The reachable workspace boundary is an envelope (article 3), and selecting configurations that reach a target is a containment query (article 6).
Physical robot joints have limits. An elbow cannot rotate a full 360 degrees. A shoulder has a restricted range. In parallel coordinates, joint limits appear as bounded ranges on each axis, with forbidden regions shaded.
When limits are imposed, the set of valid configurations shrinks. Polylines whose segments enter the forbidden region represent configurations that violate a joint constraint. The reachable workspace shrinks correspondingly, sometimes dramatically.
Drag the red boundary lines on the PC axes to adjust joint limits, or toggle limits on/off.
Figure 3. Joint limits constrain the reachable workspace. Toggle limits on/off to see how physical constraints shrink the solution space. Drag the boundary handles to change limits interactively.
Joint limits break symmetry. Without limits, reachable targets have solution families related by full rotations; with limits, some solutions become physically impossible. The forbidden shading clips the polyline family.
A 2-link arm reaching a point in 2D has exactly two degrees of freedom for a two-dimensional target. For most reachable positions, there are exactly two solutions: "elbow up" and "elbow down." The inverse kinematics problem is exactly determined.
A 3-link arm has three degrees of freedom for a two-dimensional target. There is one extra degree of freedom, and the arm is redundant. For each reachable target, there is a continuous family of solutions, not just two. The extra joint angle varies freely while the end-effector stays fixed.
Drag the target in either panel. The 2-link arm snaps to its two solutions. The 3-link arm shows a manifold of solutions.
Figure 4. Redundancy comparison. Left: a 2-link arm produces two polylines (elbow up, elbow down) for most targets. Right: a 3-link arm produces a continuous family. The extra joint angle is the null-space degree of freedom.
The 2-link arm yields two discrete polylines per target. The 3-link arm yields a 1D manifold — a band sharing the same end-effector values but varying in the third joint angle. The null-space dimension equals the band width.
For an N-dimensional configuration reaching a K-dimensional target, the solution manifold has dimension N − K. More joints means a wider band and more options for the motion planner.