Sliding mode control (SMC) is a robust nonlinear control law that first drives the tracking
error onto a designer-chosen sliding surfaces=0 (the reaching phase),
then constrains the error to slide along it to the origin (the sliding phase). Robustness comes
from a discontinuous switching term whose gain dominates the worst-case disturbance and model
uncertainty, making the closed loop insensitive to matched perturbations at the cost of
high-frequency chattering. In the space-manipulator literature SMC is used as the inner
trajectory-tracking loop that realizes a planned joint trajectory; robustness against the
base-reaction dynamic coupling is the motivation. The two cited sources sit in different regimes:
das2025 designs SMC for a free-floating SMS (uncontrolled base, the first six generalized forces
are zero), whereas zhang2020 designs a base-actuated decoupling controller for a free-flying
multi-arm robot (FB, the base force/torque, is a control input). The free-flying case is the
regime of our (fully-actuated base) system.
Linear (first-order) sliding surface on the joint-tracking error e=q−qd:
s=e˙+Λse,Λs≻0.
Control law = model-based equivalent control (keeps the trajectory on s=0 for
the nominal model) plus a robust switching term:
\qquad G_i>D_i+U_i ,$$
with the switching gain chosen to exceed the disturbance/uncertainty bound. To suppress chattering
the discontinuous $\operatorname{sgn}(\mathbf{s})$ is replaced by a boundary-layer saturation
$\operatorname{sat}(\mathbf{s}/\psi)$ (linear inside a strip of half-width $\psi$, $\pm1$ outside),
trading robustness against a small steady-state error inside the "quasi-sliding" band. Stability is
shown with $V=\tfrac12\,\mathbf{s}^{\top}\mathbf{s}$, the gain condition giving $\dot V\le0$
(reaching), after which $\mathbf{s}=\mathbf{0}$ forces $\dot{\mathbf{e}}=-\boldsymbol{\Lambda}_s\mathbf{e}$
and the error decays to the origin (sliding).
> **Notation flag.** $\boldsymbol{\Lambda}_s$ here is the *sliding-surface slope* matrix
> (das2025 writes $\Gamma$, zhang2020 writes $\Lambda$). It is **not** the
> [notation.md](../notation.md) operational-space inertia $\boldsymbol\Lambda$, nor the load-bearing
> coordinate-transform $\boldsymbol\Gamma$; the subscript $_s$ disambiguates. $\boldsymbol G$ is the
> switching-gain matrix (not the CoM-velocity map $\boldsymbol G_{v_c}$); $\psi$ is the boundary-layer
> width. These symbols are SMC-local and are not (yet) in the central registry.
## Source Support
- [das2025understanding](../sources/das2025understanding.md) — designs an SMC tracking controller for a
**free-floating** planar SMS to validate a dynamic-coupling-informed optimal trajectory; sliding
surface $\mathbf{s}=\dot{\mathbf{q}}_e+\Gamma\mathbf{q}_e$, computed-torque equivalent control plus a
saturation switching term $-K_s\operatorname{sat}(\mathbf{s}/\lambda)$, with an appendix Lyapunov
proof of reaching-phase asymptotic stability and on-surface convergence.
- [zhang2020adaptive](../sources/zhang2020adaptive.md) — uses SMC as the robustness layer of a
base-actuated decoupling controller for a **free-flying multi-arm** space robot (the generalized
force $F=[F_B;\boldsymbol\tau_M]$ includes the base drive $F_B$); contrasts CTC-based SMC with a
time-delay-estimation (TDE)-based SMC, gives the switching-gain condition $G_i>D_i+U_i$, the
boundary-layer chattering remedy, and the Lyapunov proof $\dot V\le0$. Notes that *single-arm*
free-flying control is "relatively simple" (the contribution is the *multi-arm* case) and that
SMC's high-frequency switching causes chattering.
## Related Topics
- [trajectory_tracking](trajectory_tracking.md) — SMC is deployed here specifically as a
trajectory-tracking inner loop that follows a planned joint trajectory $\boldsymbol{q}_d$.
- [lyapunov_stability](lyapunov_stability.md) — reaching and sliding phases are both certified via a
quadratic Lyapunov function; the gain condition is exactly what makes $\dot V\le0$.
- [dynamic_coupling](dynamic_coupling.md) — the disturbance SMC is hardened against: arm motion
reacts on the base, and SMC robustly tracks despite this coupling (in das2025's free-floating SMS
the base is uncontrolled and the coupling is *exploited*; in zhang2020's free-flying robot SMC
reinforces a controller that actively *decouples*/cancels the base reaction).
- [prescribed_performance_control](prescribed_performance_control.md) — an alternative robust
tracking philosophy that shapes a transient-bound funnel instead of a switching surface; both target
bounded tracking error under uncertainty.
- [ffsm_dynamics](ffsm_dynamics.md) — supplies the $(6+n)$-DOF equations of motion $\boldsymbol{M},
\boldsymbol{C}$ from which the model-based equivalent control $\boldsymbol{F}_{\mathrm{eq}}$ is built.
- [time_delay_estimation](time_delay_estimation.md) — zhang2020's alternative to a full dynamics model:
TDE estimates the lumped dynamics from delayed measurements, and SMC supplies the residual robustness.
## Open Questions
- das2025 designs SMC for the **free-floating** regime (base generalized forces = zero), while
zhang2020 already runs in a **free-flying** decoupling architecture (base force/torque $F_B$ is an
input). For our free-flying system, does das2025's surface/switching-gain construction carry over
once the base-actuation channel enters the input, and how does that change the matched-disturbance
assumption relative to zhang2020's decoupled formulation?
- Chattering is mitigated by a boundary layer at the price of a steady-state error inside the strip.
How does that residual error interact with our pointing/standoff accuracy budget versus a
continuous robust law (e.g. the super-twisting algorithm, which zhang2020 cites but does not use,
or prescribed-performance control)?
- das2025 *exploits* dynamic coupling for trajectory generation but still tracks with a generic SMC;
could a coupling-aware sliding surface tighten tracking, and is that even desirable for an actuated
base where coupling can be cancelled outright?