We present CasADi, an open-source software framework for numerical optimization. CasADi is a general-purpose tool that can be used to model and solve optimization problems with a large degree of flexibility, larger than what is associated with popular algebraic modeling languages such as AMPL, GAMS, JuMP or Pyomo. Of special interest are problems constrained by differential equations, i.e. optimal control problems. CasADi is written in self-contained C++, but is most conveniently used via full-featured interfaces to Python, MATLAB or Octave. Since its inception in late 2009, it has been used successfully for academic teaching as well as in applications from multiple fields, including process control, robotics and aerospace. This article gives an up-to-date and accessible introduction to the CasADi framework, which has undergone numerous design improvements over the last seven years.

In this paper the software environment and algorithm collection ACADO Toolkit is presented, which implements tools for automatic control and dynamic optimization. It provides a general framework for using a great variety of algorithms for direct optimal control, including model predictive control as well as state and parameter estimation. The ACADO Toolkit is implemented as a self-contained C++ code, while the object-oriented design allows for convenient coupling of existing optimization packages and for extending it with user-written optimization routines. We discuss details of the software design of the ACADO Toolkit 1.0 and describe its main software modules. Along with that we highlight a couple of algorithmic features, in particular its functionality to handle symbolic expressions. The user-friendly syntax of the ACADO Toolkit to set up optimization problems is illustrated with two tutorial examples: an optimal control and a parameter estimation problem.ACADO TOOLKIT 299 such advanced controllers. Thus, efficient and reliable optimization algorithms for performing this step-possibly on embedded hardware-are of great interest.Searching the literature, we can find a number of optimization algorithms which have been implemented for solving OCPs. We can only discuss some of the most common packages: Let us start the list with the open-source package IPOPT [1, 2], originally developed by Andreas Wachter and Larry Biegler, which implements an interior point algorithm for the optimization of large-scale differential algebraic systems. It can be combined with collocation methods for the discretization of the continous dynamic system while a filter strategy is implemented as a globalization technique. IPOPT is written in C/C++ and Fortran, but uses modeling languages such as AMPL or MATLAB in order to provide a user interface and to allow automatic differentiation.Furthermore, a MATLAB package named PROPT [3] receives more and more attention. PROPT is a commercial tool, developed by the Tomlab Optimization Inc.. PROPT solves optimal control problems based on collocation techniques, while using existing NLP solvers such as KNITRO, CONOPT, SNOPT or CPLEX. Owing to the MATLAB syntax, the package PROPT is more user-friendly than IPOPT-at the price that it is not open-source.Recently, another open-source code has been published by Brian C. Fabien [4] under the name dsoa. This package is written in C/C++ and discretizes differential algebraic systems based on implicit Runge-Kutta methods. Unfortunately, the package does only implement single-shooting methods, which is often not advisable for nonlinear OCPs. On the optimization level, sequential quadratic programming techniques are employed.Similar to dsoa, the proprietary package MUSCOD-II, originally developed by Daniel Leineweber [5], is suitable for solving OCPs. MUSCOD-II discretizes the differential algebraic systems based on backward differentiation formula (BDF) or Runge Kutta integration methods and uses Bock's direct multiple shooting [6]. Sequential quadratic pro...

Global high‐precision atmospheric Δ14CO2 records covering the last two decades are presented, and evaluated in terms of changing (radio)carbon sources and sinks, using the coarse‐grid carbon cycle model GRACE. Dedicated simulations of global trends and interhemispheric differences with respect to atmospheric CO2 as well as δ13CO2 and Δ14CO2, are shown to be in good agreement with the available observations (1940–2008). While until the 1990s the decreasing trend of Δ14CO2 was governed by equilibration of the atmospheric bomb 14C perturbation with the oceans and terrestrial biosphere, the largest perturbation today are emissions of 14C‐free fossil fuel CO2. This source presently depletes global atmospheric Δ14CO2 by 12–14‰ yr−1, which is partially compensated by 14CO2 release from the biosphere, industrial 14C emissions and natural 14C production. Fossil fuel emissions also drive the changing north–south gradient, showing lower Δ14C in the northern hemisphere only since 2002. The fossil fuel‐induced north–south (and also troposphere–stratosphere) Δ14CO2 gradient today also drives the tropospheric Δ14CO2 seasonality through variations of air mass exchange between these atmospheric compartments. Neither the observed temporal trend nor the Δ14CO2 north–south gradient may constrain global fossil fuel CO2 emissions to better than 25%, due to large uncertainties in other components of the (radio)carbon cycle.

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This paper focuses on time-optimal path-constrained trajectory planning, a subproblem in time-optimal motion planning of robot systems. Through a nonlinear change of variables, the time-optimal trajectory planning is transformed here into a convex optimal control problem with a single state. Various convexity-preserving extensions are introduced, resulting in a versatile approach for optimal trajectory planning. A direct transcription method is presented that reduces finding the globally optimal trajectory to solving a second-order cone program using robust numerical algorithms that are freely available. Validation against known examples and application to a more complex example illustrate the versatility and practicality of the new method.Index Terms-time-optimal trajectory planning, time-optimal control, convex optimal control, path tracking, trajectory planning, convex optimization, second-order cone program, direct transcription.

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The goal of this paper is to demonstrate the capacity of Model Predictive Control to generate stable walking motions without the use of predefined foot steps. Building up on well-known Model Predictive Control schemes for walking motion generation, we show that a minimal modification of these schemes allows designing an online walking motion generator which can track a given reference speed of the robot and decide automatically the foot step placement. Simulation results are proposed on the HRP-2 humanoid robot, showing a significant improvement over previous approaches.

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