Tip 0, 0, 0 mError 0 mmw 0Cond 0Reach 0 %—

Six-Axis Robot Arm: DLS IK Lab — 2D

Same kinematics as the 3D lab, shown as a side cross-section, a top-down map and the velocity ellipse. Drag in either map to move the target. Virtual model: no dynamics, no collisions.

Switch to 3D

Target (m, from the base axis)

Solver

Live follow (arm tracks the target)

Tool orientation

Planned move

Live readouts

What the 2D view shows

Orange ellipse at the tool tip: the velocity ellipse of the position Jacobian. Its semi-axes are the singular values (m/s per rad/s, drawn at 0.3 m per unit); a flat ellipse means the tool cannot move along its short axis, which is a singularity. Faint arms are ghost snapshots at 0, 33 and 66 % of the planned move; the curve is the tool-tip path.

Damping filter factors

Gain applied to an error along a singular direction: sigma/(sigma^2+lambda^2). Dashed: undamped 1/sigma. Dots: the three current singular values.

Joint angles along the planned move

Manipulability along the planned move

IK tracking error history (live)

Experiment: measured vs predicted joint step

Headless: a nearly stretched pose gets a 1 cm radial error. The measured joint-step norm of the DLS solver is compared with the singular-value prediction e*sigma_min/(sigma_min^2+lambda^2) (the error lies almost along the weakest direction).

Independent numerical checks

Controls

Click or drag in the side or top map to place the target (side map sets reach and height; top map sets X and Z).
Live follow: the arm chases the target. Switch it off and press Plan and go for a timed minimum-jerk move with ghosts.
Lambda trades accuracy for stability: near a singularity a small lambda gives huge joint steps, a large one a smooth but slightly wrong answer.

The science

Forward kinematics. T = R_y(q₁) · Tr(0,h) · R_x(q₂) · Tr(0,L₁) · R_x(q₃) · Tr(0,L₂) · R_y(q₄) · R_x(q₅) · Tr(0,L₃) · R_y(q₆) · Tr(0,L₄) with h = 0.40 m, L₁ = 0.95, L₂ = 0.85, L₃ = 0.22, L₄ = 0.30 m (model values). Maximum reach from the shoulder is 2.32 m.

Jacobian. For revolute joint i with world axis a_i and origin o_i, the linear column is a_i × (p − o_i), so v = J_v q̇.

Velocity ellipse. The unit ball of joint speeds |q̇| = 1 maps to an ellipsoid of tool velocities with semi-axes σ_i, the singular values of J_v (square roots of the eigenvalues of J_vJ_vᵀ). The 2D maps draw the projection of this ellipsoid onto the plane of the map: the 2x2 block B J_vJ_vᵀ Bᵀ, with B the two plane axes. Yoshikawa manipulability w = σ₁σ₂σ₃; condition number σ_max/σ_min.

Damped least squares. Δq = Jᵀ (J Jᵀ + λ² I)⁻¹ e (Wampler; Nakamura and Hanafusa). Along each singular direction the gain is σ/(σ²+λ²), at most 1/(2λ), instead of 1/σ for the pseudo-inverse; this is the filter-factor plot. Joint limits are enforced by clamping, which can stall IK.

Trajectories. Time law s(τ) = 10τ³ − 15τ⁴ + 6τ⁵. Joint-space interpolation is simple but the tool path curves; the Cartesian line is straight but can pass near singularities.

Simplifications. No dynamics, torques, collisions or noise; the geometry is illustrative and not a specific commercial robot. Canvas 2D only; physics module copied unchanged from the 3D version.