High Gain Observer

Definition

A high-gain observer (HGO) is a velocity estimator that reconstructs the unmeasured joint
velocity $\dot{\mathbf{q}}$ of a manipulator from position-only measurements, enabling output-feedback
control when velocity sensors are unavailable, faulty, or noisy. In yao2021adaptive
it is a linear, $\kappa$-parameterized filter whose gains scale as inverse powers of a small constant $\kappa$;
as $\kappa \to 0$ the estimation error $\mathbf{\xi }$ shrinks to an $O(\kappa )$ neighborhood of zero. The source applies it
to a free-flying (fully-actuated-base) space manipulator, so the result already lives in our regime — though the
observer itself is a generic Euler–Lagrange tool agnostic to whether the base is actuated. It supplies the velocity
estimate ${\hat{\mathbf{e}}}_2$ to the output-feedback control law in place of a measured rate.

Key Equations

Symbols per notation.md.

Second-order observer driven by the measured joint position $\mathbf{q}$ (yao2021adaptive Eq. 45):

$$\kappa {\dot{\mathbf{\zeta }}}_1={\mathbf{\zeta }}_2,\kappa {\dot{\mathbf{\zeta }}}_2=-{\delta }_1{\mathbf{\zeta }}_2-{\mathbf{\zeta }}_1+\mathbf{q},$$

with ${\mathbf{\zeta }}_1,{\mathbf{\zeta }}_2$ the observer states, $\kappa >0$ a small constant, and ${\delta }_1>0$
chosen so the characteristic polynomial $s^r+{\delta }_1{s}^{r-1}+\cdots +1$ is Hurwitz ($r=2$ here). The velocity
estimate and its bounded error are (Eqs. 46–47)

$$\widehat{\dot{\mathbf{q}}}=\frac{{\mathbf{\zeta }}_2}{\kappa },{\mathbf{\xi }}_2=\frac{{\mathbf{\zeta }}_2}{\kappa }-\dot{\mathbf{q}}=-\kappa {\mathbf{\psi }}^{\prime \prime },\Vert {\mathbf{\xi }}_2\Vert \le \kappa h_2\forall t\ge t^{\ast },$$

so the estimate ${\hat{\mathbf{e}}}_2={\mathbf{\zeta }}_2/\kappa -\mathbf{\mu }$ substitutes for the true velocity error in
the output-feedback law (Eq. 48), keeping the closed loop semi-globally uniformly ultimately bounded.

Notation flags. (i) In notation.md $\kappa$ denotes a path-curvature vector (units 1/m);
here $\kappa$ is the source’s dimensionless small-gain constant — reused source-faithfully, distinct object.
(ii) The source’s RBFNN adaptation gain ${\mathbf{\Gamma }}_i$ (Eq. 49) collides with the load-bearing
coordinate-transform matrix $\mathbf{\Gamma }$ in notation.md and is deliberately not reproduced here.

Source Support

• yao2021adaptive — primary and sole source. Designs the HGO (Eqs. 45–47, Lemma 3)
  to recover $\dot{\mathbf{q}}$ for an output-feedback adaptive controller of a free-flying space manipulator, and
  proves semi-global uniform ultimate boundedness of the resulting closed loop (Theorem 2). The HGO replaces direct
  velocity measurement, so the controller stays applicable when rate sensors are absent or corrupted.

Related Topics

• trajectory_tracking — the HGO exists to enable output-feedback trajectory tracking; its
  estimate feeds the tracking control law in place of a measured velocity.
• lyapunov_stability — boundedness of the HGO-augmented closed loop is established with a
  barrier-Lyapunov function; the $O(\kappa )$ estimation error enters the Lyapunov derivative as a vanishing perturbation.
• input_to_state_stability — the observer error acts as a bounded input disturbance to the
  control loop, the cascade structure ISS analysis addresses.
• parameter_estimation — both are state/parameter reconstruction tools; here the HGO estimates
  states (velocity) while the companion RBFNN estimates the lumped uncertainty, jointly enabling model-free control.
• sliding_mode_control — an alternative robust observer/controller family the source contrasts
  with (it cites sliding-mode disturbance-observer composite controllers as prior art).

Open Questions

• Choosing $\kappa$ small sharpens the estimate but triggers the classic HGO peaking transient and amplifies
  measurement noise; what $\kappa$ budget is admissible for a free-flying base whose actuators must reject that transient?
• The source pairs the HGO with joint-space dynamics; does it transfer cleanly to a circumcentroidal task-space
  formulation (the $\oplus$ coordinates), where the velocity to estimate is ${\mathbf{\nu }}_e^{\oplus }$ rather than $\dot{\mathbf{q}}$?
• Boundedness here is semi-global UUB under known performance bounds — what guarantee survives once chance constraints
  / CVaR-level uncertainty (the risk layer) replace the deterministic disturbance bound $\mathbf{d}$?