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Motion planning and control are two core components of the robotic systemsautonomy stack. The standard approach to combine these methodologies comprisesan offline/open-loop stage, planning, that designs a feasible and safetrajectory to follow, and an online/closed-loop stage, tracking, that correctsfor unmodeled dynamics and disturbances. Such an approach generally introducesconservativeness into the planning stage, which becomes difficult to overcomeas the model complexity increases and real-time decisions need to be made in achanging environment. This work addresses these challenges for the class ofdifferentially flat nonlinear systems by integrating planning and control intoa cohesive closed-loop task. Precisely, we develop an optimization-basedframework that aims to steer a differentially flat system to a trajectoryimplicitly defined via a constrained time-varying optimization problem. To thatend, we generalize the notion of feedback linearization, which makes non-linearsystems behave as linear systems, and develop controllers that effectivelytransform a differentially flat system into an optimization algorithm thatseeks to find the optimal solution of a (possibly time-varying) optimizationproblem. Under sufficient regularity assumptions, we prove global asymptoticconvergence for the optimization dynamics to the minimizer of the time-varyingoptimization problem. We illustrate the effectiveness of our method with twonumerical examples: a multi-robot tracking problem and a robot obstacleavoidance problem.

Closed-Loop Motion Planning for Differentially Flat Systems: A Time-Varying Optimization Framework