Impedance Control — Phase Portrait & Force-Displacement (2D)

Impedance Controller

Commanded stiffness K 1200 N/m

Damping D 70 Ns/m

Target press depth 15 mm

Environment

Wrist F/T Sensor

Contact force Fz

0.0 N

Predicted F_ss

0.0 N

Penetration

0.0 mm

EE velocity

0.0 mm/s

Shoulder torque τ₁

0.0 Nm

Elbow torque τ₂

0.0 Nm

Damping ratio ζ

0.00

Regime

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How this 2D view works

This is the same impedance-control law as the 3D arm simulator, but instead of animating a robot link it plots the controller's own state space directly. The virtual mass–spring–damper law is unchanged:

m ẍ + D(ẋ − ẋ_ref) + K(x − x_ref) = F_env
F_env = k_env · max(0, S − x)

Phase portrait (left) plots position error e = x − x_ref against velocity ẋ. Once in contact this is a damped linear system m ë + D ė + (K + k_env) e = 0, so the trajectory spirals into the origin when damping is below critical (ζ < 1, ringing/overshoot) or slides straight in when overdamped (ζ ≥ 1) — the damping ratio is computed live as ζ = D / (2√(m(K+k_env))).

Force–penetration diagram (right) plots contact force against penetration depth. The dashed line is the environment's own stiffness F = k_env·d; the steady state the controller settles onto sits where the trajectory meets the series-spring prediction F_ss = K·k_env/(K+k_env)·d, exactly as in the 3D version.

Joint torques below still come from the identical manipulator Jacobian transpose, τ = J^TF, evaluated at the arm configuration reaching the same fixed horizontal point — the same relation a real force/torque sensor and torque-controlled joints use.

K / D sliders — the virtual stiffness and damping the controller presents to the environment.
Target press depth — the commanded reference penetration d.
Soft / Medium / Rigid — the environment's own stiffness k_env.
Press / Retract — moves the reference trajectory x_ref through the surface, or back clear of it.

Phase trajectory / force path Environment stiffness line Series-spring prediction F_ss