Nonholonomic Control Research Articles

Nonholonomic Yaw Control of an Underactuated Flying Robot With Model-Based Reinforcement Learning

Nonholonomic control is a candidate to control nonlinear systems with path-dependant states. We investigate an underactuated flying micro-aerial-vehicle, the ionocraft, that requires nonholonomic control in the yaw-direction for complete attitude control. Deploying an analytical control law involves substantial engineering design and is sensitive to inaccuracy in the system model. With specific assumptions on assembly and system dynamics, we derive a Lie bracket for yaw control of the ionocraft. As a comparison to the significant engineering effort required for an analytic control law, we implement a data-driven model-based reinforcement learning yaw controller in a simulated flight task. We demonstrate that a simple model-based reinforcement learning framework can match the derived Lie bracket control (in yaw rate and chosen actions) in a few minutes of flight data, without a pre-defined dynamics function. This paper shows that learning-based approaches are useful as a tool for synthesis of nonlinear control laws previously only addressable through expert-based design.

Stabilizability in optimal control

We extend the classical concepts of sampling and Euler solutions for control systems associated to discontinuous feedbacks by considering also the corresponding costs. In particular, we introduce the notions of Sample and Euler stabilizability to a closed target set mathbf{C}with({p_{0}},W)-regulated cost, for some continuous, state-dependent function W and some constant {p_{0}}>0: it roughly means that we require the existence of a stabilizing feedback K such that all the corresponding sampling and Euler solutions starting from a point z have suitably defined finite costs, bounded above by W(z)/{p_{0}}. Then, we show how the existence of a special, semiconcave Control Lyapunov Function W, called {p_{0}}-Minimum Restraint Function, allows us to construct explicitly such a feedback K. When dynamics and Lagrangian are Lipschitz continuous in the state variable, we prove that K as above can be still obtained if there exists a {p_{0}}-Minimum Restraint Function which is merely Lipschitz continuous. An example on the stabilizability with ({p_{0}},W)-regulated cost of the nonholonomic integrator control system associated to any cost with bounded Lagrangian illustrates the results.

Nonholonomic System Modelling and Control with Single Degree of Freedom

This paper considers nonholonomic system modelling and control of a single degree of freedom. The model is based on a linear ordinary differential equation using the principles of vibrations in the area of feedback control system which is applied in many industrial applications. In this field, actual motion deviate significantly from the desired motion, and as a result of this deviation, performance, precision and accuracy of the system may not be acceptable. The problem is solved using the principles of PID and Routh-Hurwitz criterion of stability. At end the system was stable and the actual motion is the same as desired motion. The system was controllable and observable. Keywords: Nonholonomic system, PID, Routh-Hurwitz stability criterion, Controllability, Observability DOI: 10.7176/CTI/10-04 Publication date: July 31 st 2020

Ensemble observability of Bloch equations with unknown population density

We introduce in the paper a novel observability problem for a large population (in the limit, a continuum ensemble) of nonholonomic control systems with unknown population density. We address the problem by focussing on a prototype, namely, the ensemble of Bloch equations which is known for its use of describing the evolution of the bulk magnetization of a collective of non-interacting nuclear spins in a static field modulated by a radio frequency (rf) field. The dynamics of the equations are structurally identical, but show variations in Larmor dispersion and rf inhomogeneity. We assume that the initial state of any individual system (i.e., individual Bloch equation) is unknown and, moreover, the population density of these individual systems is also unknown. Furthermore, we assume that at any time, there is only one scalar measurement output at our disposal. The measurement output integrates a certain observation function, common to all individual systems, over the continuum ensemble. The observability problem we pose in the paper is thus the following: Whether one is able to use the common control input (i.e., the rf field) and the single measurement output to estimate both the initial states of the individual systems and the population density? Amongst other things, we establish a sufficient condition for the ensemble system to be observable: We show that if the common observation function is any harmonic homogeneous polynomial of positive degree, then the ensemble system is observable. The proof of the result utilizes tools from representation theory of Lie algebras. Although the results established in the paper are for the specific ensemble of Bloch equations, the approach developed along the analysis can be generalized to investigate observability of other ensemble systems with single, integrated measurement outputs.

Hamiltonian Fast Marching: A Numerical Solver for Anisotropic and Non-Holonomic Eikonal PDEs

We introduce a generalized Fast-Marching algorithm, able to compute paths globally minimizing a measure of length, defined with respect to a variety of metrics in dimension two to five. Our method applies in particular to arbitrary Riemannian metrics, and implements features such as second order accuracy, sensitivity analysis, and various stopping criteria. We also address the singular metrics associated with several non-holonomic control models, related with curvature penalization, such as the Reeds-Shepp’s car with or without reverse gear, the Euler-Mumford elastica curves, and the Dubins car. Applications to image processing and to motion planning are demonstrated.

Iterative optimal control syntheses illustrated on the Brockett integrator

In this paper, we investigate computational methods for synthesizing optimal control inputs for nonholonomic control systems. We use the Brockett integrator as a benchmark example to illustrate different aspects of the computational trajectory optimization problem. The main result of this paper is the establishment of a rather attractive iterative scheme for practically constructing optimal control inputs. The functionality and efficiency of the proposed iterative scheme are illustrated on different optimal steering problems centered around the Brockett integrator.

Design for three-dimensional stabilization control of underactuated autonomous underwater vehicles

In order to realize three-dimensional stable control of underactuated autonomous underwater vehicles (AUVs), this study analyses nonlinear hydrodynamic characteristics of AUVs based on the mathematical Taylor series. The nonholonomic control system properties of AUVs are investigated using nonholonomic system theory under three-dimensional control input. The constraint of underactuated AUVs is proved not to be integrated by the local integrability theorem, and the controllability of underactuated AUVs is verified through small-time local controllability (STLC). In order to simplify the trigonometric terms in underactuated AUV motion functions, quaternion theory is applied to transform the function to develop a continuous time-varying controller. Results of a simulation experiment show that this control law is effective in achieving three-dimensional stabilization from arbitrary initial positions.

Steering Control for a Rigid Body with two Torque Actuators using Adaptive Back Stepping

This paper presents a simple steering control algorithm for a rigid body model, which is a famous example of non-holonomic control systems with drift. The controllability Lie Algebra of a rigid body model contains Lie brackets of depth two. We propose a back-stepping-based adaptive controller design under the strict-feedback form. We analyze two cases for continuous steering. In the first case, the parameters of the model are assumed to be known while in the second case these are estimated by considering them unknown. This approach does not necessitate the conversion of the system model into a “chained form”, and thus does not rely on any special transformation techniques. The practical effectiveness of the controller is illustrated by numerical simulations and graceful stabilization.

Knife-Edge Motion on a Surface as a Nonholonomic Control Problem

This letter studies a new formulation for the kinematics of a knife-edge moving on an arbitrary smooth surface in ${\mathbb {R}^{3}}$ . The kinematics equations for a knife-edge, viewed as a rigid body, are constrained by the requirement that the knife-edge maintain contact with the surface. They describe the constrained translation of the point of contact of the knife-edge on the surface and the constrained attitude of the knife-edge as a rigid body. These equations for the knife-edge kinematics in ${\mathbb {R}}^{3}$ are expressed in a geometric form, without the use of local coordinates; they are globally defined without singularities or ambiguities. The kinematics equations can be expressed in several simplified forms and written as a drift-free nonlinear control system. Comments are made about interesting motion planning and path planning problems. The kinematics equations are specialized for two specific surfaces defined in ${ {\mathbb {R}}^{3}}$ , namely, a flat plane and the surface of a sphere. Results for the flat plane are compared with standard results obtained using local coordinates; results for the sphere, in contrast, require full attention to the 3-D geometry.

Underactuated Manipulatorの非ホロノミック制御 (高機能化を目指した非線形システム制御総合特集号-I)

Underactuated Manipulatorの非ホロノミック制御 (高機能化を目指した非線形システム制御総合特集号-I)

Control Design for a Class of Nonholonomic Systems Via Reference Vector Fields and Output Regulation

This paper presents procedural guidelines for the construction of discontinuous state feedback controllers for driftless, kinematic nonholonomic systems, with extensions to a class of dynamic nonholonomic systems with drift. Given an n-dimensional kinematic nonholonomic system subject to κ Pfaffian constraints, system states are partitioned into “leafwise” and “transverse,” based on the structure of the Pfaffian constraint matrix. A reference vector field F is defined as a function of the leafwise states only in a way that it is nonsingular everywhere except for a submanifold containing the origin. The induced decomposition of the configuration space, together with requiring the system vector field to be aligned with F, suggests choices for Lyapunov-like functions. The proposed approach recasts the original nonholonomic control problem as an output regulation problem, which although nontrivial, may admit solutions based on standard tools.

Control system of nonholonomic uniaxial wheeled module for monitoring geometry parameters of airfield pavements

We develop a control system of a nonholonomic measuring uniaxial wheeled module designed for monitoring the geometrical parameters of airfield pavements. Using a nonlinear approach, we synthesize a trajectory control subsystem based on a kinematic model of a nonholonomic unicycle locomotion control subsystem using the partial feedback linearization approach. We present the results of the numerical simulation of the synthesized control system with disturbances.

Decoupling Path Following and Velocity Profile in Vision-Guided Laser Steering

Laser surgery requires accurate following of a path defined by the surgeon, while the velocity on this path is dependent on the laser-tissue interaction. Therefore, path following and velocity profile control must be decoupled. In this paper, nonholonomic control of the unicycle model is used to implement velocity-independent visual path following for laser surgery. The proposed controller was tested in simulation, as well as experimentally in several conditions of use: different initial velocities (step input, successive step inputs, sinusoidal inputs), optimized/nonoptimized gains, time-varying path (simulating a patient breathing), and complex curves with curvatures. Thereby, experiments at 587 Hz (frames/s) show an average accuracy lower than 0.22 pixels (~10 μm) with a standard deviation of 0.55 pixels (~25 μm) path following, and a relative velocity distortion of less than 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-6</sup> %.

Tracking control of a non-holonomic wheeled mobile robot using improved compound cosine function neural networks

The purpose of this paper is to propose an improved compound cosine function neural network (NN) controller to improve the tracking control performance of the compound cosine function NN controller for a non-holonomic mobile robot with nonlinear disturbances. Since the hidden layer neuron functions chosen in the three-layer network structure can affect the learning and control capability of the NN controller, the hidden layer neuron functions in the improved NN are composed of the compound functions of the cosine function and the arcsine function combined with a unipolar sigmoid function. The main advantages of this improved NN controller are able to effectively improve the rapid real-time control capability and control accuracy in the non-holonomic mobile robot control system with nonlinear disturbances and to have the simple learning algorithm of the NN controller. The effectiveness of the improved NN controller is demonstrated by simulation experiments.

Analysis the brachistochronic motion of a mechanical system with nonlinear nonholonomic constraint

This paper analyzes the problem of brachistochronic planar motion of a mechanical system with nonlinear nonholonomic constraint. The nonholonomic system is represented by two Chaplygin blades of negligible dimensions, which impose nonlinear constraint in the form of perpendicularity of velocities. The brachistrochronic planar motion is considered, with specified initial and terminal positions, at unchanged value of mechanical energy during motion. Differential equations of motion, where the reactions of nonholonomic constraints and control forces figure, are obtained on the basis of general theorems of mechanics. Here, this is more convenient to use than some other methods of analytical mechanics applied to nonholonomic mechanical systems, where a subsequent physical interpretation of the multipliers of constraints is required. The formulated brachistochrone problem, with adequately chosen quantities of state, is solved as simple a task of optimal control as possible in this case by applying the Pontryagin maximum principle. The corresponding two-point boundary value problem of the system of ordinary nonlinear differential equations is obtained, which has to be numerically solved. Numerical procedure for solving the two-point boundary value problem is performed by the method of shooting. On the basis of thus obtained brachistochronic motion, the active control forces, along with the reactions of nonholonomic constraints, are defined. Using the Coulomb friction laws, a minimum required value of the coefficient of sliding friction is defined, so that the considered system is moving in accordance with nonholonomic bilateral constraints.

10.5937/fmet1403199r = Analysis of the minimum required coefficient of sliding friction at brachistochronic motion of a nonholonomic mechanical system

The paper analyzes the problem of brachistochronic motion of a nonholonomic mechanical system, using an example of a simple car model. The system moves between two default positions at an unaltered value of the mechanical energy during motion. Differential equations of motion, containing the reaction of nonholonomic constraints and control forces, are obtained on the basis of general theorems of dynamics. Here, this is more appropriate than some other methods of analytical mechanics applied to nonholonomic systems, where the provision of a subsequent physical interpretation of the multipliers of constraints is required to solve this problem. By the appropriate choice of the parameters of state as simple a task of optimal control as possible is obtained in this case, which is solved by the application of the Pontryagin maximum principle. Numerical solution of the two-point boundary value problem is obtained by the method of shooting. Based on the thus acquired brachistochronic motion, the active control forces are determined as well as the reaction of constraints. Using the Coulomb laws of friction sliding, the minimum value of the coefficient of friction is determined to avoid car skidding at the points of contact with the ground.

G151022 観測ノイズを有するX4フライヤーの非ホロノミック制御([G151-02]ロボティクスメカトロニクス部門,一般セッション(2))

Recently,VTOL aerial robots are expected to be used for collecting any information at the time of a disaster.The aerial robot called X4-Flyer can control at least three attitudes and one position by changing the rotor speed of four rotors.However,the control performance of a controller for the X4-flyer is deteriorated due to measurement noises generated by the rotational vibration of the rotors.Therefore,it is important to reduce the effect of such measurement noises on the controlled system.In this paper,when using a nonholonomic control method based on the Lyapunov stability theory to design a basic controller for the X4-Flyer,it aims at reducing the deterioration of the controller performance,which is caused by the influence of measurement noises,by processing such noises with an extended Kalman filter.The effectiveness of the proposed method is checked through numerical simulations.

A nonholonomic control method for stabilizing an X4-AUV

A nonholonomic control method is considered for stabilizing all attitudes and positions (x, y, or z) of an underactuated X4 autonomous underwater vehicle (AUV) with four thrusters and six degrees of freedom (DOF), in which the positions are stabilized according to the Lyapunov stability theory. A dynamic model is first derived, and then a sequential nonlinear control strategy is implemented for the X4-AUV which is composed of translational and rotational subsystems. A controller for the translational subsystem stabilizes one position out of the x-, y-, and z-coordinates, whereas controllers for the rotational subsystems generate the desired roll, pitch, and yaw angles. Thus, the rotational controllers stabilize all the attitudes of the X4-AUV at the desired (x-, y-, or z-) position of the vehicle. Some numerical simulations are conducted to demonstrate the effectiveness of the proposed controllers.

Nonholonomic quantum control

In this paper, we review an open-loop evolution control method, called nonholonomic control, based on the alternate application of only two physical perturbations for timings which play the role of adjustable control parameters. We present the algorithm which allows one to explicitly compute the pulse sequence achieving any arbitrarily prescribed unitary evolution in a nonholonomic system, i.e. a system subject to two physical perturbations which, together with the natural Hamiltonian of the system, span the entire Lie algebra . We moreover expose two extensions of our method to open quantum systems which, respectively, aim at preserving the information stored in the system and safely processing this information. The first is based on a generalization of the quantum Zeno effect, while the second is inspired by decoupling pulse techniques. The most important feature of the methods presented here is their universality: they indeed do not rely on any specific assumption on the system, which in particular is not bound to be a collection of two-level systems, or the error model considered, as is usually the case in the literature. Numerical and physical applications of our techniques are also provided.

Robust stabilization for nonholonomic systems with state delay and nonlinear drifts

This paper presents a control strategy for stabilization of the nonholonomic control systems with strongly nonlinear drifts and state delay. Applying a novel Lyapunov functional and backstepping recursive method, the design of robust nonlinear state feedback controllers is proposed, which can guarantee the stability of the closed-loop systems. Finally, a numerical example is provided to show the effectiveness of the method.