  Recent strides in nonlinear model predictive control (NMPC) underscore a
dependence on numerical advancements to efficiently and accurately solve
large-scale problems. Given the substantial number of variables characterizing
typical whole-body optimal control (OC) problems - often numbering in the
thousands - exploiting the sparse structure of the numerical problem becomes
crucial to meet computational demands, typically in the range of a few
milliseconds. Addressing the linear-quadratic regulator (LQR) problem is a
fundamental building block for computing Newton or Sequential Quadratic
Programming (SQP) steps in direct optimal control methods. This paper
concentrates on equality-constrained problems featuring implicit system
dynamics and dual regularization, a characteristic of advanced interiorpoint or
augmented Lagrangian solvers. Here, we introduce a parallel algorithm for
solving an LQR problem with dual regularization. Leveraging a rewriting of the
LQR recursion through block elimination, we first enhanced the efficiency of
the serial algorithm and then subsequently generalized it to handle parametric
problems. This extension enables us to split decision variables and solve
multiple subproblems concurrently. Our algorithm is implemented in our
nonlinear numerical optimal control library ALIGATOR. It showcases improved
performance over previous serial formulations and we validate its efficacy by
deploying it in the model predictive control of a real quadruped robot.
