This paper presents a preview steering control algorithm and its closed-loop system analysis and experimental validation for accurate, smooth, and computationally inexpensive path tracking of automated vehicles. The path tracking issue is formulated as an optimal control problem with dynamic disturbance, i.e., the future road curvature. A discrete-time preview controller is then designed on the top of a linear augmented error system, in which the disturbances within a finite preview window are augmented as part of the state vector. The obtained optimal steering control law is in an analytic form and consists of two parts: 1) a feedback control responding to tracking errors and 2) a feedforward control dealing with the future road curvatures. The designed control’s nature, capacity, computation load, and underlying mechanism are revealed by the analysis of system responses in the time domain and the frequency domain, theoretical steady-state error, and comparison with the model predictive control (MPC). The algorithm was implemented on an automated vehicle platform, a hybrid Lincoln MKZ. The experimental and simulation results are then presented to demonstrate the improved performance in tracking accuracy, steering smoothness, and computational efficiency compared to the MPC and the full-state feedback control.

Design, Analysis, and Experiments of Preview Path Tracking Control for Autonomous Vehicles

We show that artificial neural networks with rectifier units as activation functions can exactly represent the piecewise affine function that results from the formulation of model predictive control (MPC) of linear time-invariant systems. The choice of deep neural networks is particularly interesting as they can represent exponentially many more affine regions compared to networks with only one hidden layer. We provide theoretical bounds on the minimum number of hidden layers and neurons per layer that a neural network should have to exactly represent a given MPC law. The proposed approach has a strong potential as an approximation method of predictive control laws, leading to a better approximation quality and significantly smaller memory requirements than previous approaches, as we illustrate via simulation examples. We also suggest different alternatives to correct or quantify the approximation error. Since the online evaluation of neural networks is extremely simple, the approximated controllers can be deployed on low-power embedded devices with small storage capacity, enabling the implementation of advanced decision-making strategies for complex cyber-physical systems with limited computing capabilities.

Efficient Representation and Approximation of Model Predictive Control Laws via Deep Learning

Tracking control of robot manipulators is a fundamental and significant problem in robotic industry. As a conventional solution, the Jacobian-matrix-pseudo-inverse (JMPI) method suffers from two major limitations: one is the requirement on known information of the robot model such as parameter and structure; the other is the position error accumulation phenomenon caused by the open-loop nature. To overcome such two limitations, this paper proposes a novel Jacobian-matrix-adaption (JMA) method for the tracking control of robot manipulators via the zeroing dynamics. Unlike existing works requiring the information of the known robot model, the proposed JMA method uses only the input-output information to control the robot with unknown model. The solution based on the JMA method transforms the internal, implicit, and unmeasurable model information to the external, explicit, and measurable input-output information. Moreover, simulation studies including comparisons and tests substantiate the efficacy and superiority of the proposed JMA method for the tracking control of robot manipulators subject to unknown models.

Tracking Control of Robot Manipulators with Unknown Models: A Jacobian-Matrix-Adaption Method

We consider the tracking of geometric paths in output spaces of nonlinear systems subject to input and state constraints without pre-specified timing requirements, commonly referred to as constrained output path-following problems. Specifically, we propose a predictive control approach to constrained path-following problems with and without velocity assignments. We present sufficient convergence conditions based on terminal regions and end penalties. Furthermore, we analyze the geometric nature of constrained output path-following problems and thereby provide insight into the computation of suitable terminal control laws and terminal regions. We draw upon an example from robotics to illustrate our findings.

Nonlinear Model Predictive Control for Constrained Output Path Following

This paper presents the acados software package, a collection of solvers for fast embedded optimization intended for fast embedded applications. Its interfaces to higher-level languages make it useful for quickly designing an optimization-based control algorithm by putting together different algorithmic components that can be readily connected and interchanged. Since the core of acados is written on top of a high-performance linear algebra library, we do not sacrifice computational performance. Thus, we aim to provide both flexibility and performance through modularity, without the need to rely on automatic code generation, which facilitates maintainability and extensibility. The main features of acados are: efficient optimal control algorithms targeting embedded devices implemented in C, linear algebra based on the high-performance BLASFEO library, user-friendly interfaces to Matlab and Python, and compatibility with the modeling language of CasADi. acados is free and open-source software released under the permissive BSD 2-Clause license.

acados: a modular open-source framework for fast embedded optimal control

Many robotic applications, such as milling, gluing, or high precision measurements, require the precise following of a predefined geometric path. We investigate the real-time feasible implementation of model predictive path-following control for an industrial robot. We consider constrained output path following with and without reference speed assignment. Finally, we present the results of an implementation of the proposed model predictive path-following controller on a KUKA LWR IV robot.

Implementation of Nonlinear Model Predictive Path-Following Control for an Industrial Robot

Model predictive control (MPC) is an optimization-based strategy for high-performance control that is attracting increasing interest. While MPC requires the online solution of an optimization problem, its ability to handle multivariable systems and constraints makes it a very powerful control strategy specially for MPC of embedded systems, which have an ever increasing amount of sensing and computation capabilities. We argue that the implementation of MPC on field programmable gate arrays (FPGAs) using automatic tools is nowadays possible, achieving cost-effective successful applications on fast or resource-constrained systems. The main burden for the implementation of MPC on FPGAs is the challenging design of the necessary algorithms. We outline an approach to achieve a software-supported optimized implementation of MPC on FPGAs using high-level synthesis tools and automatic code generation. The proposed strategy exploits the arithmetic operations necessaries to solve optimization problems to tailor an FPGA design, which allows a tradeoff between energy, memory requirements, cost, and achievable speed. We show the capabilities and the simplicity of use of the proposed methodology on two different examples and illustrate its advantages over a microcontroller implementation.

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Optimized FPGA Implementation of Model Predictive Control for Embedded Systems Using High-Level Synthesis Tool

Predictive control, embedded cyberphysical systems and systems of systems – A perspective

Implementing linear model predictive controllers in embedded systems with limited computational resources is still challenging. Recently, several code generation tools have been developed that produce highly efficient library-free optimization algorithms. We present a tool that focuses on controller performance and hardware with low computational resources. The underlying optimization algorithm has been explicitly developed for real-time embedded applications, and is based on an augmented Lagrangian method together with Nesterov's gradient method. The tool provides offline methods that allow the generation of online controllers that have low computational requirements and quickly reach good performance. We demonstrate the capabilities of the software, and the performance of the generated controllers with two examples.

μAO-MPC: A free code generation tool for embedded real-time linear model predictive control

We discuss implementation related aspects of model predictive control schemes on embedded platforms. Ex-emplarily, we focus on fast gradient methods and present results from an implementation on a low-cost microcontroller. We show that input quantization in actuators should be exploited in order to determine a suboptimality level of the online optimization that requires a low number of algorithm iterations and might not significantly degrade the performance of the real system. As a case study we consider a Segway-like robot, modeled by a linear time-invariant system with 8 states and 2 inputs subject to box input constraints. The test system runs with a sampling period of 4 ms and uses a horizons up to 20 steps in a hard real-time system with limited CPU time and memory.

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Pablo Zometa

Papers

Implementation of Nonlinear Model Predictive Path-Following Control for an Industrial Robot

Optimized FPGA Implementation of Model Predictive Control for Embedded Systems Using High-Level Synthesis Tool

Predictive control, embedded cyberphysical systems and systems of systems – A perspective

μAO-MPC: A free code generation tool for embedded real-time linear model predictive control

Implementation aspects of model predictive control for embedded systems