Christian Kirches

qpOASES: a parametric active-set algorithm for quadratic programming

Many practical applications lead to optimization problems that can either be stated as quadratic programming (QP) problems or require the solution of QP problems on a lower algorithmic level. One relatively recent approach to solve QP problems are parametric active-set methods that are based on tracing the solution along a linear homotopy between a QP problem with known solution and the QP problem to be solved. This approach seems to make them particularly suited for applications where a-priori information can be used to speed-up the QP solution or where high solution accuracy is required. In this paper we describe the open-source C++ software package qpOASES, which implements a parametric active-set method in a reliable and efficient way. Numerical tests show that qpOASES can outperform other popular academic and commercial QP solvers on small- to medium-scale convex test examples of the Maros-Mészáros QP collection. Moreover, various interfaces to third-party software packages make it easy to use, even on embedded computer hardware. Finally, we describe how qpOASES can be used to compute critical points of nonconvex QP problems.

Mixed-integer nonlinear optimization

Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems. Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques. Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations. We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP.

A Reactive Walking Pattern Generator Based on Nonlinear Model Predictive Control

The contribution of this work is to show that real-time nonlinear model predictive control (NMPC) can be implemented on position controlled humanoid robots. Following the idea of “walking without thinking,” we propose a walking pattern generator that takes into account simultaneously the position and orientation of the feet. A requirement for an application in real-world scenarios is the avoidance of obstacles. Therefore, this letter shows an extension of the pattern generator that directly considers the avoidance of convex obstacles. The algorithm uses the whole-body dynamics to correct the center of mass trajectory of the underlying simplified model. The pattern generator runs in real-time on the embedded hardware of the humanoid robot HRP2 and experiments demonstrate the increase in performance with the correction.

Efficient Numerics for Nonlinear Model Predictive Control

We review a closely connected family of algorithmic approaches for fast and real–time capable nonlinear model predictive control (NMPC) of dynamic processes described by ordinary differential equations or index-1 differential-algebraic equations. Focusing on active–set based algorithms, we present emerging ideas on adaptive updates of the local quadratic subproblems (QPs) in a multi–level scheme. Structure exploiting approaches for the solution of these QP subproblems are the workhorses of any fast active–set NMPC method. We present linear algebra tailored to the QP block structures that act both as a preprocessing and as block structured factorization methods.

Block-structured quadratic programming for the direct multiple shooting method for optimal control

In this contribution, we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, for example, from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved using sequential quadratic programming methods. We review the classical condensing algorithm that preprocesses the large but structured quadratic programs (QPs) to obtain small but dense ones. We show that this approach leaves room for improvement when applied in conjunction with outer convexification. To this end, we present a new complementary condensing algorithm for QPs with many controls. This algorithm is based on a hybrid null-space range-space approach to exploit the block structure of the QPs that is due to direct multiple shooting. An assessment of the theoretical run-time complexity reveals significant advantages of the proposed algorithm. We give a detailed account on the required number of floating point operations, depending on the process dimensions. Finally, we demonstrate the merit of the new complementary condensing approach by comparing the behaviour of both methods for a vehicle control problem in which the integer gear decision is convexified.

A factorization with update procedures for a KKT matrix arising in direct optimal control

Quadratic programs obtained for optimal control problems of dynamic or discrete-time processes usually involve highly block structured Hessian and constraints matrices, to be exploited by efficient numerical methods. In interior point methods, this is elegantly achieved by the widespread availability of advanced sparse symmetric indefinite factorization codes. For active set methods, however, conventional dense matrix techniques suffer from the need to update base matrices in every active set iteration, thereby loosing the sparsity structure after a few updates. This contribution presents a new factorization of a KKT matrix arising in active set methods for optimal control. It fully respects the block structure without any fill-in. For this factorization, matrix updates are derived for all cases of active set changes. This allows for the design of a highly efficient block structured active set method for optimal control and model predictive control problems with long horizons or many control parameters.

Optimal Control of Formula 1 Race Cars in a VDrift Based Virtual Environment

Abstract Control of autonomous vehicles and providing recommendations to drivers in real time are challenging tasks from an algorithmic point of view. To include realistic effects, such as nonlinear tire dynamics, at least medium-sized mathematical models need to be considered. Yet, fast feedback is of utmost importance. Existing Nonlinear Model Predictive Control (NMPC) algorithms need to be enhanced to comply with these two contradictory requirements. As the testing of algorithms in an automatic driving context is cumbersome and expensive, we propose a virtual testbed for NMPC of driving cars. We use the open source race simulator VDrift as virtual real world, in which algorithms need to cope with the mismatch between the detailed physical model in the simulator and a coarser approximative model used for NMPC. We present the general framework of this virtual environment and an optimal control problem based on a medium-sized ordinary differential equation model and a generic and flexible parameterization of the track constraint. We discuss one possible algorithmic approach to the task of minimum time driving including gear shifts and give preliminary open loop numerical results for a Porsche on Germanys Formula One racing circuit Hockenheimring. This can be used as a reference against which other (closed loop) solutions can be compared in the future.

Combinatorial integral approximation

We are interested in structures and efficient methods for mixed-integer nonlinear programs (MINLP) that arise from a first discretize, then optimize approach to time-dependent mixed-integer optimal control problems (MIOCPs). In this study we focus on combinatorial constraints, in particular on restrictions on the number of switches on a fixed time grid. We propose a novel approach that is based on a decomposition of the MINLP into a NLP and a MILP. We discuss the relation of the MILP solution to the MINLP solution and formulate bounds for the gap between the two, depending on Lipschitz constants and the control discretization grid size. The MILP solution can also be used for an efficient initialization of the MINLP solution process. The speedup of the solution of the MILP compared to the MINLP solution is considerable already for general purpose MILP solvers. We analyze the structure of the MILP that takes switching constraints into account and propose a tailored Branch and Bound strategy that outperforms cplex on a numerical case study and hence further improves efficiency of our novel method.

Optimal control for whole-body motion generation using center-of-mass dynamics for predefined multi-contact configurations

Multi-contact motion generation is an important problem in humanoid robotics because it generalizes bipedal locomotion and thus expands the functional range of humanoid robots. In this paper, we propose a complete solution to compute a fully-dynamic multi-contact motion of a humanoid robot. We decompose the motion generation by computing first a dynamically-consistent trajectory of the center of mass of the robot and finding then the whole-body movement following this trajectory. A simplified dynamic model of the humanoid is used to find optimal contact forces as well as a kinematic feasible center-of-mass trajectory from a predefined series of contacts. We demonstrate the capabilities of the approach by making the real humanoid robot platform HRP-2 climb stairs with the use of a handrail. The experimental study also shows that utilization of the handrail lowers the power consumption of the robot by 25% compared to a motion, where only the feet are used.

Fast Nonlinear Model Predictive Control with an Application in Automotive Engineering

Although nonlinear model predictive control has become a well-established control approach, its application to time-critical systems requiring fast feedback is still a major computational challenge. In this article we investigate a new multi-level iteration scheme based on theory and algorithmic ideas from [2], and extending the idea of real-time iterations as presented in [4]. This novel approach takes into account the natural hierarchy of different time scales inherent in the dynamic model. Applications from aerodynamics and chemical engineering have been successfully treated. In this contribution we apply the investigated multi-level iteration scheme to fast optimal control of a vehicle and discuss the computational performance of the scheme.