Overview#
The Thrust Allocator distributes generalized control forces $\tau \in \mathbb{R}^n$ to the AUV’s actuators as control inputs $u \in \mathbb{R}^r$. For linear systems this reduces to $\tau = Bu$, where $B$ is the input matrix. The individual control inputs $u_i$ are passed downstream to the thruster interface.
This module is part of Vortex NTNU’s autonomous underwater vehicle stack.
Solvers#
Pseudoinverse Allocator#
Follows the unconstrained weighted least-squares formulation from Fossen (2021, Eq. 11.27):
$$ J = \min_{f_e} \; f_e^\top W_f f_e \qquad \text{s.t.} \qquad \tau - T_e f_e = 0 $$Solving yields the generalized pseudoinverse (Fossen Eq. 11.35):
$$ T_w^+ = W_f^{-1} T_e^\top \left( T_e W_f^{-1} T_e^\top \right)^{-1} $$Since the Orca drone uses 8 identical thrusters, there is no practical reason to weight actuators differently. With $W_f = I$, the solution simplifies to the right Moore–Penrose pseudoinverse (Fossen Eq. 11.36) Our new drone, however, will utilize 8 thrusters and they will have different maximum and minimum thrust in the forward and backward direction. To keep optimality a more advanced method could be utilized
$$ T^+ = T_e^\top (T_e T_e^\top)^{-1} $$Constrained QP Allocator#
Formulated as a quadratic program following Fossen (2021, Eq. 11.38). The original formulation includes a load-balancing parameter $\bar{f}$, but for our use case different maneuvers require some thrusters to work hard while others rest — so load balancing does more harm than good.
The implemented formulation drops the load-balancing term:
$$ \begin{aligned} J &= \min_{f_e,\, s} \; \left( f_e^\top W_f f_e + s^\top Q s \right) \\ &\text{s.t.} \quad T_e f_e = \tau + s, \qquad f_{\min} \le f_e \le f_{\max} \end{aligned} $$This retains thrust limits and soft constraint handling via the slack vector $s$ while avoiding unnecessary force redistribution.
Notation#
All formulations follow the notation of Fossen (2021), Ch. 11:
- $\tau \in \mathbb{R}^n$ — desired generalized force
- $T_e$ — extended thruster configuration matrix
- $f_e$ — extended force vector
- $W_f \succeq 0$ — weighting matrix on the extended force vector
- $Q \succeq 0$ — weighting matrix on the slack vector $s$
- $f_{\min}, f_{\max}$ — lower and upper bounds on $f_e$
Acknowledgements#
The thrust allocation formulations are based on:
Thor I. Fossen, Handbook of Marine Craft Hydrodynamics and Motion Control, 2nd Edition, Wiley, 2021 — Chapter 11.