  The Sequential Linear Quadratic (SLQ) algorithm is a continuous-time variant
of the well-known Differential Dynamic Programming (DDP) technique with a
Gauss-Newton Hessian approximation. This family of methods has gained
popularity in the robotics community due to its efficiency in solving complex
trajectory optimization problems. However, one major drawback of DDP-based
formulations is their inability to properly incorporate path constraints. In
this paper, we address this issue by devising a constrained SLQ algorithm that
handles a mixture of constraints with a previously implemented projection
technique and a new augmented-Lagrangian approach. By providing an appropriate
multiplier update law, and by solving a single inner and outer loop iteration,
we are able to retrieve suboptimal solutions at rates suitable for real-time
model-predictive control applications. We particularly focus on the
inequality-constrained case, where three augmented-Lagrangian penalty functions
are introduced, along with their corresponding multiplier update rules. These
are then benchmarked against a relaxed log-barrier formulation in a cart-pole
swing up example, an obstacle-avoidance task, and an object-pushing task with a
quadrupedal mobile manipulator.
