Barrier Lyapunov Function

Definition

A barrier Lyapunov function (BLF) is a Lyapunov-like function that is positive-definite, $C^1$ on an
open set containing the origin, and grows without bound as its argument approaches a prescribed
boundary. In control design, the BLF is built on a constrained error signal so that keeping the BLF
bounded along closed-loop trajectories certifies the constraint is never violated. Yao et al. (2021) use
an asymmetric, time-varying logarithmic BLF on the generalized-coordinate position-tracking error
${\mathbf{e}}_1=\mathbf{q}-{\mathbf{q}}_d$ of an attitude-controlled free-flying space manipulator, where
$\mathbf{q}\in {\mathbb{R}}^n$ is the generalized position — the base attitude plus the arm joint angles
(Yao §2, Eq. 1) — so the error is confined inside time-varying prescribed-performance envelopes
$[-{\mathbf{k}}_a(t),{\mathbf{k}}_b(t)]$ for all time, which the authors invoke to keep the base and arm from
colliding. The regime is explicitly free-flying (attitude-actuated base), and the controller is posed in
the generalized-coordinate (configuration) space of $\mathbf{q}$, not task space — a contrast our
circumcentroidal task-space formulation must keep in mind.

Key Equations

Symbols per notation.md.

Note: ${\mathbf{e}}_1$ (joint position error), the constraint bounds $k_{ai},k_{bi}$, the performance
function ${\delta }_i(t)$, and the backstepping gain ${\mathbf{K}}_1$ below are source-local (Yao 2021) and are
not in notation.md; ${\mathbf{K}}_1$ here is a backstepping gain, distinct from the stiffness
$\breve{\mathbf{K}}$ in notation.md. They are reproduced source-faithfully.

Logarithmic-barrier inequality (the property that makes the BLF a barrier — Yao Lemma 2, after Tee/Ge):
for $|x|<k_a$,

$\ln \left(\frac{k_a^2}{k_a^2-x^2}\right)\le \frac{x^2}{k_a^2-x^2}.$

Asymmetric time-varying BLF on the joint error (Yao Eq. 16), with the sign-selector
$p(i)=1$ if $e_{1i}>0$ else $0$:

$V_1=\frac{1}{2}{\sum }_{i=1}^n[p(i)\ln \frac{k_{bi}^2(t)}{k_{bi}^2(t)-e_{1i}^2(t)}+(1-p(i)\left)\ln \frac{k_{ai}^2(t)}{k_{ai}^2(t)-e_{1i}^2(t)}\right].$

As $|e_{1i}|\to k_{ai}$ or $k_{bi}$ the corresponding logarithm diverges, so a bounded $V_1$ implies the
prescribed-performance bound (Yao Eq. 14) is never breached:

\delta_i(t)=(e_{1i,0}-e_{1i,\infty})e^{-\gamma_i t}+e_{1i,\infty}.$$ ## Source Support - [yao2021adaptive](../sources/yao2021adaptive.md) — primary. Designs an asymmetric time-varying logarithmic BLF (Eq. 16) inside a backstepping controller for an attitude-controlled free-flying space manipulator; pairs it with an RBFNN for lumped uncertainty and a high-gain observer for output feedback, proving semi-global uniform ultimate boundedness while the BLF certifies the joint errors stay within the prescribed bounds. The barrier property rests on the logarithmic inequality (Lemma 2, citing Tee/Ge/Ren). ## Related Topics - [lyapunov_stability](lyapunov_stability.md) — the BLF is a Lyapunov-like function; boundedness of $V_1$ (rather than $V_1\to0$) is what certifies constraint satisfaction, so it specializes Lyapunov analysis. - [prescribed_performance_control](prescribed_performance_control.md) — the BLF is one mechanism to enforce prescribed-performance envelopes; Yao contrasts it with the error-transformation route (Bechlioulis–Rovithakis). - [trajectory_tracking](trajectory_tracking.md) — the BLF is embedded in the tracking controller to bound the position-tracking error $\mathbf{e}_1=\mathbf{q}-\mathbf{q}_d$ throughout the maneuver. - [high_gain_observer](high_gain_observer.md) — in Yao's output-feedback variant the BLF controller is fed velocity estimates from a high-gain observer, so the barrier guarantee holds without velocity sensors. - [task_space_error_dynamics](task_space_error_dynamics.md) — Yao constrains the *generalized-coordinate* (base-attitude + joint) error $\mathbf{e}_1$; our formulation tracks *task*-space (circumcentroidal EE) error, so applying a BLF here would constrain $\tilde{\boldsymbol x}_e$ instead. ## Open Questions - Yao constrains the generalized-coordinate error $\mathbf{e}_1$ (base attitude + joint angles); does the same logarithmic BLF transfer to a **task-space** error $\tilde{\boldsymbol x}_e$ (e.g. a versine pointing error) without the constraint becoming conservative or vacuous near a circumcentroidal singularity? - The BLF requires the *initial* error to start inside the envelope ($-\mathbf{k}_a(0)<\mathbf{e}_1(0)<\mathbf{k}_b(0)$). How does this feasibility precondition interact with our startup ramp and the steady-state cruise-lag error floor $\tilde{\boldsymbol x}_{ss}$, which is nonzero by construction? - Yao's free-flying base is attitude-controlled, and the BLF acts on the generalized-coordinate error $\mathbf{e}_1$ (base **attitude** + arm joints, but not base **translation**); for our fully-actuated 6-DOF base, can a BLF simultaneously bound base-pose (including translation) and EE-pose errors, and does the asymmetric form remain well-posed under the base–arm dynamic coupling?