Cartesian Impedance Control

Definition

Cartesian impedance control regulates the dynamic relationship between end-effector motion and
external force rather than tracking a pose directly: the controller makes the closed loop behave as
a prescribed mass-spring-damper in task coordinates, with symmetric positive-definite desired
stiffness, damping, and (optionally) inertia matrices about a virtual equilibrium pose ${\mathbf{x}}_d$ (ott2008cartesian, §3.1). It is built on Khatib’s
operational space formulation: the joint-space rigid-body model is
rewritten in $m$ task coordinates $\mathbf{x}=f(\mathbf{q})$, and the impedance law is then
mapped back to joint torques through ${\mathbf{J}}^{\top }$. The source is fixed-base (rigid
terrestrial manipulators, DLR lightweight robots) — it assumes neither a space base nor base
reaction; transfer to our free-flying system is via the operational-space / coupled-dynamics
treatment, not assumed.

Key Equations

Symbols per notation.md.

Desired closed-loop impedance about the virtual equilibrium. Ott’s general law (Eq. 3.8) leaves the
desired inertia ${\mathbf{\Lambda }}_d$ a free SPD matrix, generally $\ne$ the actual task inertia; the
form below is the no-inertia-shaping closed loop (Ott Eq. 3.16 with ${\mathbf{\Lambda }}_d=\mathbf{\Lambda }$,
Eq. 3.15), in the canonical EE error ${\tilde{\mathbf{x}}}_e$ with EE stiffness/damping ${\mathbf{K}}_e,{\mathbf{D}}_e$ and $\mathbf{\Lambda }$ the Khatib operational-space task inertia — Ott writes these
${\mathbf{K}}_d,{\mathbf{D}}_d$ (the Coriolis term $\mathbf{\mu }{\dot{\tilde{\mathbf{x}}}}_e$ that
Eq. 3.16 carries alongside ${\mathbf{D}}_e$ is suppressed here):

$\mathbf{\Lambda }{\ddot{\tilde{\mathbf{x}}}}_e+{\mathbf{D}}_e{\dot{\tilde{\mathbf{x}}}}_e+{\mathbf{K}}_e{\tilde{\mathbf{x}}}_e={\mathbf{w}}_e,{\tilde{\mathbf{x}}}_e={\mathbf{x}}_e-{\mathbf{x}}_{ed}$

Joint-torque realization without inertia shaping (Ott Eq. 3.18), which needs no force measurement
— here $\mathbf{\mu }$ is Ott’s task-space Coriolis matrix (not in notation.md) and $\mathbf{g}$
the gravity term ($\mathbf{g}=0$ in orbit):

$\mathbf{\tau }=\mathbf{g}(\mathbf{q})+{\mathbf{J}}^{\top }\left(\mathbf{\Lambda }{\ddot{\mathbf{x}}}_{ed}+\mathbf{\mu }{\dot{\mathbf{x}}}_{ed}-{\mathbf{K}}_e{\tilde{\mathbf{x}}}_e-{\mathbf{D}}_e{\dot{\tilde{\mathbf{x}}}}_e\right)$

In the regulation case (${\dot{\mathbf{x}}}_{ed}=0$) the closed loop is a passive map
from external wrench ${\mathbf{w}}_e$ to EE velocity ${\dot{\mathbf{x}}}_e$ (Ott Prop. 3.5), the
stability anchor of the scheme.

Source Support

• ott2008cartesian — primary; Ch. 3 derives the rigid-body Cartesian
  impedance law, the no-inertia-shaping simplification, generalized-eigenvalue damping design
  (§3.3), passivity (Prop. 3.5), singularity-avoidance via superposition of impedances (§3.4), and
  unit-quaternion rotational stiffness to dodge representation singularities (§3.5.2).

Related Topics

• operational_space_control — the Khatib formulation the impedance law
  is built on; supplies the task-space inertia $\mathbf{\Lambda }=(\mathbf{J}{\mathbf{M}}^{-1}{\mathbf{J}}^{\top }{)}^{-1}$ and the ${\mathbf{J}}^{\top }$ torque map.
• task_space_error_dynamics — the mass-spring-damper error equation
  (Eq. 3.8) is exactly the prescribed task-space error dynamics this controller imposes.
• trajectory_tracking — with a time-varying virtual equilibrium
  ${\mathbf{x}}_{ed}(t)$ the impedance controller becomes a compliant tracker (the regulation case
  ${\dot{\mathbf{x}}}_{ed}=0$ recovers pure compliance control).
• cartesian_path_planning — supplies the virtual-equilibrium reference
  path ${\mathbf{x}}_{ed}(t)$ in task space that the impedance controller is compliant about.
• null_space_control — for a redundant arm, impedance torques live in the
  task channel; null-space torques (e.g. singularity avoidance, Ott §3.4) are superposed without
  perturbing the Cartesian behavior.
• kinematic_singularity — near a body-Jacobian singularity the desired
  impedance is distorted and the manipulator can get stuck; Ott §3.4 superposes a second impedance
  that pushes away from singular configurations.

Open Questions

• Ott assumes a fixed base with invertible $\mathbf{J}$; for our free-flying base the relevant
  Jacobian is the circumcentroidal ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ and the
  task inertia is the reduced $\breve{\mathbf{M}}$ — does the §3.5 quaternion-stiffness/passivity
  argument carry over once base reaction and the $\mathbf{\Gamma }$ coordinate transform enter?
• The no-inertia-shaping law (Eq. 3.18) avoids force feedback by leaving the apparent inertia
  unshaped; with a 6-DOF actuated base and arm coupling, is the natural coupled inertia an acceptable
  impedance, or must $\mathbf{\Lambda }$ be shaped (requiring an EE wrench estimate)?
• Singularity avoidance by impedance superposition (§3.4) can create a spurious potential minimum near
  the singularity — how does this interact with our ${\sigma }_G$ threshold cascade and impedance-derate
  ramp $\gamma ({\sigma }_G)$?