arXiv subjects

Moritz Diehl

Publications and source records attributed to Moritz Diehl.

At least 73 records · Page 4Linked to original sources

arXiv2023

The Time-Freezing Reformulation for Numerical Optimal Control of Complementarity Lagrangian Systems with State Jumps

This paper introduces a novel time-freezing reformulation and numerical methods for optimal control of complementarity Lagrangian systems (CLS) with state jumps. We cover the difficult case when the system evolves on the boundary of the dynamic's feasible set after the state jump. In nonsmooth mechanics, this corresponds to inelastic impacts. The main idea of the time-freezing reformulation is to introduce a clock state and an auxiliary dynamical system whose trajectory endpoints satisfy the state jump law. When the auxiliary system is active, the clock state is not evolving, hence by taking only the parts of the trajectory when the clock state was active, we can recover the original solution. The resulting time-freezing system is a Filippov system that has jump discontinuities only in the first time derivative instead of the trajectory itself. This enables one to use the recently proposed Finite Elements with Switch Detection [Nurkanovic et al., 2022], which makes high accuracy numerical optimal control of CLS with impacts and friction possible. We detail how to recover the solution of the original system and show how to select appropriate auxiliary dynamics. The theoretical findings are illustrated on a nontrivial numerical optimal control example of a hopping one-legged robot.

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arXiv2023

Collision-free Motion Planning for Mobile Robots by Zero-order Robust Optimization-based MPC

This paper presents an implementation of robust model predictive control (MPC) for collision-free reference trajectory tracking for mobile robots. The presented approach considers the robot motion to be subject to process noise bounded by ellipsoidal sets. In order to efficiently handle the evolution of the disturbance ellipsoids within the MPC, the zero-order robust optimization (zoRO) scheme is applied. The idea is to fix the disturbance ellipsoids within one optimization iteration and solve the problem repeatedly with updated disturbance ellipsoid trajectories. The zero-order approach is suboptimal in general. However, we show that it does not impair convergence to the reference trajectory in the absence of obstacles. The experiments on an industrial mobile robot prototype demonstrate the performance of the controller.

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arXiv2023

FESD-J: Finite Elements with Switch Detection for Numerical Optimal Control of Rigid Bodies with Impacts and Coulomb Friction

The Finite Elements with Switch Detection (FESD) is a high-accuracy method for the numerical simulation and solution of optimal control problems subject to discontinuous ODEs. In this article, we extend the FESD method [Nurkanović et al., 2022] to the dynamic equations of multiple rigid bodies that exhibit state jumps due to impacts and Coulomb friction. This new method is referred to as FESD with Jumps (FESD-J). Starting from the standard Runge-Kutta equations, we let the integration step sizes be degrees of freedom. Additional constraints are introduced to ensure exact switch detection and to remove spurious degrees of freedom if no switches occur. Moreover, at the boundaries of each finite element, we impose the impact equations in their complementarity form, at both the position and velocity level. They compute the normal and tangential impulses in case of contact making. Otherwise, they are reduced to the continuity conditions for the velocities. FESD-J treats multiple contacts, where each contact can have a different coefficient of restitution and friction. All methods introduced in this paper are implemented in the open-source software package NOSNOC. We illustrate the use of FESD-J in both simulation and optimal control examples.

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arXiv2023

Direct Collocation for Numerical Optimal Control of Second-Order ODE

Mechanical systems are usually modeled by second-order Ordinary Differential Equations (ODE) which take the form $\ddot{q} = f(t, q, \dot{q})$. While simulation methods tailored to these equations have been studied, using them in direct optimal control methods is rare. Indeed, the standard approach is to perform a state augmentation, adding the velocities to the state. The main drawback of this approach is that the number of decision variables is doubled, which could harm the performance of the resulting optimization problem. In this paper, we present an approach tailored to second-order ODE. We compare it with the standard one, both on theoretical aspects and in a numerical example. Notably, we show that the tailored formulation is likely to improve the performance of a direct collocation method, for solving optimal control problems with second-order ODE of the more restrictive form $\ddot{q} = f(t, q)$.

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arXiv2023

Direct Collocation Methods for Trajectory Optimization in Constrained Robotic Systems

Direct collocation methods are powerful tools to solve trajectory optimization problems in robotics. While their resulting trajectories tend to be dynamically accurate, they may also present large kinematic errors in the case of constrained mechanical systems, i.e., those whose state coordinates are subject to holonomic or nonholonomic constraints, like loop-closure or rolling-contact constraints. These constraints confine the robot trajectories to an implicitly-defined manifold, which complicates the computation of accurate solutions. Discretization errors inherent to the transcription of the problem easily make the trajectories drift away from this manifold, which results in physically inconsistent motions that are difficult to track with a controller. This paper reviews existing methods to deal with this problem and proposes new ones to overcome their limitations. Current approaches either disregard the kinematic constraints (which leads to drift accumulation) or modify the system dynamics to keep the trajectory close to the manifold (which adds artificial forces or energy dissipation to the system). The methods we propose, in contrast, achieve full drift elimination on the discrete trajectory, or even along the continuous one, without artificial modifications of the system dynamics. We illustrate and compare the methods using various examples of different complexity.

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arXiv2023

Local Convergence Behaviour of Generalized Gauss-Newton Multiple Shooting, Single Shooting and Differential Dynamic Programming

We revisit three classical numerical methods for solving unconstrained optimal control problems - multiple shooting, single shooting, and differential dynamic programming - and examine their local convergence behaviour. In particular, we show that all three methods converge with the same linear rate if a Gauss-Newton (GN), or more general a Generalized Gauss-Newton (GGN), Hessian approximation is used, which is the case in widely used implementations such as iLQR.

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arXiv2023

Imitation Learning from Nonlinear MPC via the Exact Q-Loss and its Gauss-Newton Approximation

This work presents a novel loss function for learning nonlinear Model Predictive Control policies via Imitation Learning. Standard approaches to Imitation Learning neglect information about the expert and generally adopt a loss function based on the distance between expert and learned controls. In this work, we present a loss based on the Q-function directly embedding the performance objectives and constraint satisfaction of the associated Optimal Control Problem (OCP). However, training a Neural Network with the Q-loss requires solving the associated OCP for each new sample. To alleviate the computational burden, we derive a second Q-loss based on the Gauss-Newton approximation of the OCP resulting in a faster training time. We validate our losses against Behavioral Cloning, the standard approach to Imitation Learning, on the control of a nonlinear system with constraints. The final results show that the Q-function-based losses significantly reduce the amount of constraint violations while achieving comparable or better closed-loop costs.

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arXiv2023

An Efficient Method for the Joint Estimation of System Parameters and Noise Covariances for Linear Time-Variant Systems

We present an optimization-based method for the joint estimation of system parameters and noise covariances of linear time-variant systems. Given measured data, this method maximizes the likelihood of the parameters. We solve the optimization problem of interest via a novel structure-exploiting solver. We present the advantages of the proposed approach over commonly used methods in the framework of Moving Horizon Estimation. Finally, we show the performance of the method through numerical simulations on a realistic example of a thermal system. In this example, the method can successfully estimate the model parameters in a short computational time.

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arXiv2023

A dual-control effect preserving formulation for nonlinear output-feedback stochastic model predictive control with constraints

We propose a formulation for approximate constrained nonlinear output-feedback stochastic model predictive control. Starting from the ideal but intractable stochastic optimal control problem (OCP), which involves the optimization over output-dependent policies, we use linearization with respect to the uncertainty to derive a tractable approximation which includes knowledge of the output model. This allows us to compute the expected value for the outer functions of the OCP exactly. Crucially, the dual control effect is preserved by this approximation. In consequence, the resulting controller is aware of how the choice of inputs affects the information available in the future which in turn influences subsequent controls.

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arXiv2022

Frenet-Cartesian Model Representations for Automotive Obstacle Avoidance within Nonlinear MPC

In recent years, nonlinear model predictive control (NMPC) has been extensively used for solving automotive motion control and planning tasks. In order to formulate the NMPC problem, different coordinate systems can be used with different advantages. We propose and compare formulations for the NMPC related optimization problem, involving a Cartesian and a Frenet coordinate frame (CCF/ FCF) in a single nonlinear program (NLP). We specify costs and collision avoidance constraints in the more advantageous coordinate frame, derive appropriate formulations and compare different obstacle constraints. With this approach, we exploit the simpler formulation of opponent vehicle constraints in the CCF, as well as road aligned costs and constraints related to the FCF. Comparisons to other approaches in a simulation framework highlight the advantages of the proposed approaches.

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arXiv2022

A Hierarchical Approach for Strategic Motion Planning in Autonomous Racing

We present an approach for safe trajectory planning, where a strategic task related to autonomous racing is learned sample-efficient within a simulation environment. A high-level policy, represented as a neural network, outputs a reward specification that is used within the cost function of a parametric nonlinear model predictive controller (NMPC). By including constraints and vehicle kinematics in the NLP, we are able to guarantee safe and feasible trajectories related to the used model. Compared to classical reinforcement learning (RL), our approach restricts the exploration to safe trajectories, starts with a good prior performance and yields full trajectories that can be passed to a tracking lowest-level controller. We do not address the lowest-level controller in this work and assume perfect tracking of feasible trajectories. We show the superior performance of our algorithm on simulated racing tasks that include high-level decision making. The vehicle learns to efficiently overtake slower vehicles and to avoid getting overtaken by blocking faster vehicles.

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arXiv2022

A Feasible Sequential Linear Programming Algorithm with Application to Time-Optimal Path Planning Problems

In this paper, we propose a Feasible Sequential Linear Programming (FSLP) algorithm applied to time-optimal control problems (TOCP) obtained through direct multiple shooting discretization. This method is motivated by TOCP with nonlinear constraints which arise in motion planning of mechatronic systems. The algorithm applies a trust-region globalization strategy ensuring global convergence. For fully determined problems our algorithm provides locally quadratic convergence. Moreover, the algorithm keeps all iterates feasible enabling early termination at suboptimal, feasible solutions. This additional feasibility is achieved by an efficient iterative strategy using evaluations of constraints, i.e., zero-order information. Convergence of the feasibility iterations can be enforced by reduction of the trust-region radius. These feasibility iterations maintain feasibility for general Nonlinear Programs (NLP). Therefore, the algorithm is applicable to general NLPs. We demonstrate our algorithm's efficiency and the feasibility update strategy on a TOCP of an overhead crane motion planning simulation case.

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arXiv2022

LCQPow -- A Solver for Linear Complementarity Quadratic Programs

In this paper we introduce an open-source software package written in C++ for efficiently finding solutions to quadratic programming problems with linear complementarity constraints. These problems arise in a wide range of applications in engineering and economics, and they are challenging to solve due to their structural violation of standard constraint qualifications, and highly nonconvex, nonsmooth feasible sets. This work extends a previously presented algorithm based on a sequential convex programming approach applied to a standard penalty reformulation. We examine the behavior of local convergence and introduce new algorithmic features. Competitive performance profiles are presented in comparison to state-of-the-art solvers and solution variants in both existing and new benchmarks.

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arXiv2022

Vertical Airborne Wind Energy Farms with High Power Density per Ground Area based on Multi-Aircraft Systems

This paper proposes and simulates vertical airborne wind energy (AWE) farms based on multi-aircraft systems with high power density (PD) per ground area. These farms consist of many independently ground located systems that are flying at the same inclination angle, but with different tether lengths, such that all aircraft fly in a large planar elliptical area that is vertical to the tethers. The individual systems are assigned non-overlapping flight cylinders depending on the wind direction. Detailed calculations that take into account Betz' limit, assuming a cubically averaged wind power density of 7 m/s, give a potential yearly average PD of 43 MW/km$^2$. A conventional wind farm with typical packing density would yield a PD of 2.4 MW/km$^2$ in the same wind field. More refined simulations using optimal control result in a more modest PD of 6 MW/km$^2$ for practically recommended flight trajectories. This PD can already be achieved with small-scale aircraft with a wing span of 5.5 m. The simulations additionally show that the achievable PD is more than an order of magnitude higher than for a single-aircraft AWE system with the same wing span.

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arXiv2022

An Implicit and Explicit Dual Model Predictive Control Formulation for a Steel Recycling Process

We present a formulation for both implicit and explicit dual model predictive control for a steel recycling process. The process consists in the production of new steel by choosing a combination of several different steel scraps with unknown pollutant content. The pollutant content can only be measured after a scrap combination is molten, allowing for inference on the pollutants in the different scrap heaps. The production cost should be minimized while ensuring high quality of the product through constraining the maximum amount of pollutant. The dual control formulation allows to achieve the optimal explore-exploit trade-off between uncertainty reduction and cost minimization for the examined problem. Specifically, the dual effect is obtained by considering the dependence of the future pollutant uncertainties on the scrap selection in the predictions. The implicit formulation promotes uncertainty reduction indirectly via the impact of active constraints on the objective, while the explicit formulation adds a heuristic cost on uncertainty to encourage active exploration. We compare the formulations by numerical simulations of a simplified but representative industrial steel recycling process. The results demonstrate the superiority of the two dual formulations with respect to a robustified but non-dual formulation.

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arXiv2022

Fast integrators with sensitivity propagation for use in CasADi

Efficient integrators with sensitivity propagation are an essential ingredient for the numerical solution of optimal control problems. This paper gives an overview on the acados integrators, their Python interface and presents a workflow that allows using them with their sensitivities within a nonlinear programming (NLP) solver interfaced by CasADi. The implementation is discussed, demonstrated and provided as open-source software. The computation times of the proposed integrator and its sensitivity computation are compared to the native CasADi collocation integrator, CVODES and IDAS on different examples. A speedup of one order of magnitude for simulation and of up to three orders of magnitude for the forward sensitivity propagation is shown for an airborne wind energy system model.

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arXiv2022

NOSNOC: A Software Package for Numerical Optimal Control of Nonsmooth Systems

This letter introduces the NOnSmooth Numerical Optimal Control (NOSNOC) open-source software package. It is a modular MATLAB tool based on CasADi and IPOPT for numerically solving Optimal Control Problems (OCP) with piecewise smooth systems (PSS). The tool supports: 1) automatic reformulation of systems with state jumps into PSS (via the time-freezing reformulation [Nurkanović et al., 2021]) and of PSS into computationally more convenient forms, 2) automatic discretization of the OCP via, e.g., the recently introduced Finite Elements with Switch Detection [Nurkanović et al., 2022] which enables high accuracy optimal control and simulation of PSS, 3) solution methods for the resulting discrete-time OCP. The nonsmooth discrete-time OCP are solved with techniques of continuous optimization in a homotopy procedure, without the use of integer variables. This enables the treatment of a broad class of nonsmooth systems in a unified way. Two tutorial examples are given. A benchmark shows that NOSNOC provides both faster and more accurate solutions than conventional approaches, including mixed-integer formulations.

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arXiv2022

Continuous Optimization for Control of Hybrid Systems with Hysteresis via Time-Freezing

This article regards numerical optimal control of a class of hybrid systems with hysteresis using solely techniques from nonlinear optimization, without any integer variables. Hysteresis is a rate independent memory effect which often results in severe nonsmoothness in the dynamics. These systems are not simply Piecewise Smooth Systems (PSS); they are a more complicated form of hybrid systems. We introduce a time-freezing reformulation which transforms these systems into a PSS. From the theoretical side, this reformulation opens the door to study systems with hysteresis via the rich tools developed for Filippov systems. From the practical side, it enables the use of the recently developed Finite Elements with Switch Detection [Nurkanovic et al., 2022], which makes high accuracy numerical optimal control of hybrid systems with hysteresis possible. We provide a time optimal control problem example and compare our approach to mixed-integer formulations from the literature.

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