A Family of Iterative Gauss-Newton Shooting Methods for Nonlinear Optimal Control

This paper introduces a family of iterative algorithms for unconstrained nonlinear optimal control. We generalize the well-known iLQR algorithm to different multiple-shooting variants, combining advantages like straight-forward initialization and a closed-loop forward integration. All algorithms have similar computational complexity, i.e. linear complexity in the time horizon, and can be derived in the same computational framework. We compare the full-step variants of our algorithms and present several simulation examples, including a high-dimensional underactuated robot subject to contact switches. Simulation results show that our multiple-shooting algorithms can achieve faster convergence, better local contraction rates and much shorter runtimes than classical iLQR, which makes them a superior choice for nonlinear model predictive control applications.Comment: 8 page

A Semismooth Predictor Corrector Method for Real-Time Constrained Parametric Optimization with Applications in Model Predictive Control

Real-time optimization problems are ubiquitous in control and estimation, and are typically parameterized by incoming measurement data and/or operator commands. This paper proposes solving parameterized constrained nonlinear programs using a semismooth predictor-corrector (SSPC) method. Nonlinear complementarity functions are used to reformulate the first order necessary conditions of the optimization problem into a parameterized non-smooth root-finding problem. Starting from an approximate solution, a semismooth Euler-Newton algorithm is proposed for tracking the trajectory of the primal-dual solution as the parameter varies over time. Active set changes are naturally handled by the SSPC method, which only requires the solution of linear systems of equations. The paper establishes conditions under which the solution trajectories of the root-finding problem are well behaved and provides sufficient conditions for ensuring boundedness of the tracking error. Numerical case studies featuring the application of the SSPC method to nonlinear model predictive control are reported and demonstrate the advantages of the proposed method

Chance-Constrained Trajectory Optimization for Safe Exploration and Learning of Nonlinear Systems

Learning-based control algorithms require data collection with abundant supervision for training. Safe exploration algorithms ensure the safety of this data collection process even when only partial knowledge is available. We present a new approach for optimal motion planning with safe exploration that integrates chance-constrained stochastic optimal control with dynamics learning and feedback control. We derive an iterative convex optimization algorithm that solves an \underline{Info}rmation-cost \underline{S}tochastic \underline{N}onlinear \underline{O}ptimal \underline{C}ontrol problem (Info-SNOC). The optimization objective encodes both optimal performance and exploration for learning, and the safety is incorporated as distributionally robust chance constraints. The dynamics are predicted from a robust regression model that is learned from data. The Info-SNOC algorithm is used to compute a sub-optimal pool of safe motion plans that aid in exploration for learning unknown residual dynamics under safety constraints. A stable feedback controller is used to execute the motion plan and collect data for model learning. We prove the safety of rollout from our exploration method and reduction in uncertainty over epochs, thereby guaranteeing the consistency of our learning method. We validate the effectiveness of Info-SNOC by designing and implementing a pool of safe trajectories for a planar robot. We demonstrate that our approach has higher success rate in ensuring safety when compared to a deterministic trajectory optimization approach.Comment: Submitted to RA-L 2020, review-

Comparative evaluation of approaches in T.4.1-4.3 and working definition of adaptive module

The goal of this deliverable is two-fold: (1) to present and compare different approaches towards learning and encoding movements us- ing dynamical systems that have been developed by the AMARSi partners (in the past during the first 6 months of the project), and (2) to analyze their suitability to be used as adaptive modules, i.e. as building blocks for the complete architecture that will be devel- oped in the project. The document presents a total of eight approaches, in two groups: modules for discrete movements (i.e. with a clear goal where the movement stops) and for rhythmic movements (i.e. which exhibit periodicity). The basic formulation of each approach is presented together with some illustrative simulation results. Key character- istics such as the type of dynamical behavior, learning algorithm, generalization properties, stability analysis are then discussed for each approach. We then make a comparative analysis of the different approaches by comparing these characteristics and discussing their suitability for the AMARSi project

Multi-Level Iteration Schemes with Adaptive Level Choice for Nonlinear Model Predictive Control

In this thesis we develop the Multi-Level Iteration schemes (MLI), a numerical method for Nonlinear Model Predictive Control (NMPC) where the dynamical models are described by ordinary differential equations. The method is based on Direct Multiple Shooting for the discretization of the optimal control problems to be solved in each sample. The arising parametric nonlinear problems are solved approximately by setting up a generalized tangential predictor in a preparation phase. This generalized tangential predictor is given by a quadratic program (QP), which implicitly defines a piecewise affine linear feedback law. The feedback law is then evaluated in a feedback phase by solving the QP for the current state estimate as soon as it becomes known to the controller. The method developed in this thesis yields significant computational savings by updating the matrix and vector data of the tangential predictor in a hierarchy of four levels. The lowest level performs no updates and just calculates the feedback for a new initial state estimate. The second level updates the QP constraint functions and approximates the QP gradient. The third level updates the QP constraint functions and calculates the exact QP gradient. The fourth level evaluates all matrix and vector data of the QP. Feedback schemes are then assembled by choosing a level for each sample. This yields a successive update of the piecewise affine linear feedback law that is implicitly defined by the generalized tangential predictor. We present and discuss four strategies for data communication between the levels in a scheme and we describe how schemes with fixed level choices can be assembled in practice. We give local convergence theory for each level type holding its own set of primal-dual variables for fixed initial values, and discuss existing convergence theory for the case of a closed-loop process. We outline a modification of the levels that yields additional computational savings. For the adaptive choice of the levels at runtime, we develop two contraction-based criteria to decide whether the currently used linearization remains valid and use them in an algorithm to decide which level to employ for the next sample. Furthermore, we propose a criterion applicable to online estimation. The criterion provides additional information for the level decision for the next sample. Focusing on the second lowest level, we propose an efficient algorithm for suboptimal NMPC. For the presented algorithmic approaches, we describe structure exploitation in the form of tailored condensing, outline the Online Active Set Strategy as an efficient way to solve the quadratic subproblems and extend the method to linear least-squares problems. We develop iterative matrix-free methods for one contraction-based criterion, which estimates the spectral radius of the iteration matrix. We describe three application fields where MLI provides significant computational savings compared to state-of-the-art numerical methods for NMPC. For both fixed and adaptive MLI schemes, we carry out extensive numerical testings for challenging nonlinear test problems and compare the performance of MLI to a state-of-the-art numerical method for NMPC. The schemes obtained by adaptive MLI are computationally much cheaper while showing comparable performance. By construction, the adaptive MLI allows giving feedback with a much higher frequency, which significantly improves controller performance for the considered test problems. To perform the numerical experiments, we have implemented the proposed method within a MATLAB(R) based software called MLI, which makes use of a software package for the automatic derivative generation of first and higher order for the solution of the dynamic model as well as objective and constraint functions, which performs structure exploitation by condensing, and which efficiently solves the parametric quadratic subproblems by using a software package that provides an implementation of the Online Active Set Strategy