  Robot programming tools ranging from inverse kinematics (IK) to model
predictive control (MPC) are most often described as constrained optimization
problems. Even though there are currently many commercially-available
second-order solvers, robotics literature recently focused on efficient
implementations and improvements over these solvers for real-time robotic
applications. However, most often, these implementations stay problem-specific
and are not easy to access or implement, or do not exploit the geometric aspect
of the robotics problems. In this work, we propose to solve these problems
using a fast, easy-to-implement first-order method that fully exploits the
geometric constraints via Euclidean projections, called Augmented Lagrangian
Spectral Projected Gradient Descent (ALSPG). We show that 1. using projections
instead of full constraints and gradients improves the performance of the
solver and 2. ALSPG stays competitive to the standard second-order methods such
as iLQR in the unconstrained case. We showcase these results with IK and motion
planning problems on simulated examples and with an MPC problem on a 7-axis
manipulator experiment.
