Nonlinear MPC

Definition

Nonlinear Model Predictive Control (NMPC) is the model-predictive control strategy in which the
prediction model is the full nonlinear plant — here the satellite–manipulator equations of motion —
used directly without any linearization. At each sampling instant the controller solves a finite-horizon
optimal control problem (OCP) over the predicted states, applies only the first control of the optimized
sequence, then shifts the horizon forward (the receding-horizon strategy). In rybus2017control NMPC is the
real-time tracking module that drives the end-effector along a reference trajectory produced by a separate
offline trajectory-optimization module; the source assumes the free-floating regime (uncontrolled base,
arm reaction compensated by base motion), modelled with a generalized-coordinate state of dimension $n+6$,
and notes the system is nonholonomic. This is not our free-flying regime — see Open Questions.

Key Equations

Symbols per notation.md.

At each instant the controller solves, over a prediction horizon, a least-squares tracking OCP

$$\underset{\mathbf{u}(\cdot )}{\min }{\int }_t^{t+T_p}\Vert {\mathbf{x}}_e(\sigma )-{\mathbf{x}}_{e,\mathrm{ref}}(\sigma ){\Vert }_{\mathbf{W}}^2\mathrm{d}\sigma \text{s.t.}{\ddot{\mathbf{q}}}_p={\mathbf{M}}^{-1}\left(\mathbf{Q}-\mathbf{C}{\dot{\mathbf{q}}}_p\right),$$

with the extended tracking state ${\mathbf{x}}_e=[{\mathbf{v}}_s{\mathbf{\omega }}_s\dot{\mathbf{\theta }}{\mathbf{r}}_s{\mathbf{\Theta }}_s\mathbf{\theta }{\mathbf{r}}_{ee}{]}^{\top }$ and the end-effector position ${\mathbf{r}}_{ee}$ appended so it can be penalized directly. Only the first control of the optimized sequence is applied, then the horizon recedes. The control torque is split as

$$\mathbf{u}={\mathbf{u}}_{ref}+{\mathbf{u}}_{contr}({\mathbf{e}}_p,{\mathbf{e}}_v),{\mathbf{e}}_p={\mathbf{r}}_{ee}-({\mathbf{r}}_{ee}{)}_{ref},{\mathbf{e}}_v={\mathbf{v}}_{ee}-({\mathbf{v}}_{ee}{)}_{ref},$$

though the source’s NMPC uses the full nonlinear model and discards ${\mathbf{u}}_{ref}$, relying on ${\mathbf{u}}_{contr}$ alone.

Notation note: $\mathbf{M},\mathbf{C}$ match notation.md (coupled inertia / Coriolis); $\mathbf{W}$ (LSQ weighting
matrix), $\mathbf{Q}$ (generalized forces), $T_p$ (prediction-horizon length, $1\mathrm{s}$ over $20$ intervals in the source),
and the source’s free-floating state symbols ${\mathbf{v}}_s,{\mathbf{\omega }}_s,\mathbf{\theta },{\mathbf{r}}_s,{\mathbf{\Theta }}_s,{\mathbf{r}}_{ee}$ are
not in notation.md (this page is general/technique, not the circumcentroidal FFSM model) — reproduced source-faithfully, not canonicalized.

Source Support

• rybus2017control — primary: builds a two-module control system (offline
  trajectory optimization + real-time NMPC) for a free-floating planar satellite–manipulator; NMPC uses
  the full nonlinear $n+6$-DOF model with no linearization, an ACADO least-squares objective, and outperforms
  a Dynamic-Jacobian-inverse controller and MSAC under disturbances and parameter mismatch. Notes that prior
  space-manipulator MPC assumed a fixed base and that high real-time cost is NMPC’s main drawback.

Related Topics

• model_predictive_control — the parent technique; NMPC is the nonlinear
  (no-linearization) instance of it.
• receding_horizon_control — the implicit feedback mechanism NMPC uses:
  re-solve, apply first move, shift the horizon.
• trajectory_tracking — NMPC’s role here is end-effector trajectory tracking; the
  OCP cost penalizes the tracking error ${\mathbf{x}}_e-{\mathbf{x}}_{e,\mathrm{ref}}$.
• trajectory_optimization — supplies the offline reference trajectory that NMPC
  then realizes online; the two modules are complementary (global plan vs. real-time correction).
• sequential_convex_programming — an alternative way to solve the nonlinear
  OCP (convexify-and-iterate) versus the direct nonlinear-program solve used here via ACADO.

Open Questions

• The source’s NMPC assumes a free-floating base (state folds in CoM position and base attitude, system
  is nonholonomic). For our free-flying (fully-actuated 6-DOF base) system the base is directly actuated
  and momentum is not conserved — does the same OCP structure carry over, and does the nonholonomic
  characterization even apply once the base is independently commanded?
• The source flags high real-time computational cost as NMPC’s major disadvantage (especially for $>4$-DOF
  manipulators with their generalized-coordinate model). Is full-nonlinear NMPC tractable in real time for a
  redundant free-flying arm, or is convexification (sequential_convex_programming)
  or feedback linearization needed?
• Stability is asserted empirically (“no unstable behaviour observed”) rather than proven; the source itself
  calls for an analytical stability analysis. What terminal cost / constraint would certify recursive
  feasibility and stability for the free-flying case?