Lev Rapoport

Motion control for a wheeled robot following a curvilinear path

A control synthesis problem for planar motion of a wheeled robot with regard to the steering gear dynamics is considered. The control goal is to bring the robot to a given curvilinear path and to stabilize its motion along the path. The trajectory is assumed to be an arbitrary parameterized smooth curve satisfying some additional curvature constraints. A change of variables is found by means of which the system of differential equations governing controlled motion of the robot reduces to the form that admits feedback linearization. A control law is synthesized for an arbitrary target path with regard to phase and control constraints. The form of the boundary manifold and the phase portrait of the system for the case of the straight target trajectory are studied. Results of numerical experiments are presented.

Smoothing curvature of trajectories constructed by noisy measurements in path planning problems for wheeled robots

A path planning problem for a wheeled robot is considered. The problem consists in constructing a trajectory that approximates a given ordered sequence of points on the plane and satisfies certain smoothness requirements and curvature constraints. Such a problem arises, for example, when it is required to follow in an automated mode a path stored as a discrete set of points measured in the course of the first passage of this path in a manual mode. Due to errors inherent in the data points, the shape of the curve approximating the desired path may turn out inappropriate or even unacceptable from the control standpoint. The shape of the curve can be improved by applying the so-called fairing, which consists in moving the original data points with the aim to minimize some functional. Adequate small variations of the data points (within the measurement error) preserve the proximity of the resulting path to the original data points and, at the same time, may considerably improve its shape. In the paper, a new global fairing method for improving shape of curves consisting of elementary B-splines is proposed. The improvement is achieved through minimization of jumps of the spline third derivative. The problem of finding desired variations reduces to solving a quadratic programming problem with simple constraints. The discussion is illustrated by numerical examples.

Stabilization Problem for a Wheeled Robot Following a Curvilinear Path on Uneven Terrain

A control synthesis problem for a wheeled robot moving on uneven terrain is studied. The terrain is assumed to be described by a sufficiently smooth function that does not vary too much at distances of the order of the platform size, which makes it possible to employ a planar robot model. The terrain model is not a priori known, and the information on the local terrain configuration is made available for the robot through measuring its pitch and roll angles. The control goal is to bring the robot to a given curvilinear path and to stabilize robot’s motion along it. A change of variables is found by means of which the system of differential equations governing controlled motion of the robot reduces to the form that admits feedback linearization. A numerical example presented demonstrates advantages of the synthesized control compared to that derived without regard to the terrain unevenness. It is shown that the latter is generally not capable of stabilizing robot’s motion with a desired accuracy.

Canonical representation of the path following problem for wheeled robots

For the problem of stabilizing motion of an n-dimensional nonholonomic wheeled system along a prescribed path, the concept of a canonical representation of the equations of motion is introduced. The latter is defined to be a representation that can easily be reduced to a linear system in stabilizable variables by means of an appropriate nonlinear feedback. In the canonical representation, the path following problem is formulated as that of stabilizing the zero solution of an (n−1)-dimensional subsystem of the canonical system. It is shown that, by changing the independent variable, the construction of the canonical representation reduces to finding the normal form of a stationary affine system. The canonical representation is shown to be not unique and is determined by the choice of the independent variable. Three changes of variables known from the literature, which were earlier used for synthesis of stabilizing controls for wheeled robot models described by the third- and fourth-order systems of equations, are shown to be canonical ones and can be generalized to the n-dimensional case. Advantages and disadvantages of the linearizing control laws obtained by means of these changes of variables are discussed.

Global Energy Fairing of B-Spline Curves in Path Planning Problems

The paper is concerned with path planning for mobile robots. Specifically, the discussion is related to the following problem: Given an ordered sequence of points on the plane, construct a path that fits these points and satisfies certain smoothness requirements. These requirements may be different in different problems and imply basically that the constructed path is to be realizable. Such a problem arises, e.g., when it is required to follow in an automated mode a path stored as a discrete set of points, which, e.g., were collected by a GPS receiver installed on a car when it followed this path for the first time. Due to errors inherent in the data points, the shape of the curve approximating the desired path turns out often inappropriate. The shape of the curve can be improved by applying the so-called fairing, which consists in moving the original data points with the aim to minimize some fairness criterion. Adequate small variations of the data points preserve the proximity of the resulting path to the original data points and make it fairer. In the paper, a new global fairing method is proposed. It reduces the problem of constructing a fair cubic B-spline curve to solving a quadratic programming problem with simple constraints. The fairing criterion is based on minimizing jumps of the spline third derivative. The discussion is illustrated by numerical examples of fairing two actual paths constructed by data points collected by a GPS/GLONASS receiver mounted on a moving vehicle.Copyright

Ellipsoidal Approximations of Invariant Sets in Stabilization Problem for a Wheeled Robot Following a Curvilinear Path

A stabilization problem for a wheeled robot following a curvilinear target path is studied. In [1], a method for constructing invariant ellipsoids—quadratic approximations of the attraction domains for the target trajectory under a given control law—was developed. A basic result of that study is a theorem by means of which construction of the invariant ellipsoids reduces to solving a system of linear matrix inequalities (LMIs) and checking a scalar inequality. This paper is a sequel to work [1] and is devoted to practical implementation of the results obtained in that paper. It is discussed how to select the parameters in terms of which the theorem is formulated. An algorithm is developed that, for a given value of maximal deviation from the target trajectory, constructs an invariant ellipsoid of as large volume as possible.Copyright

Canonical representation of a nonstationary path following problem

The path following problem for kinematic models of wheeled robots governed by nonlinear nonstationary affine systems with scalar control is considered. The concept of a canonical representation for this problem is introduced. The path following problem in a canonical form is stated as that of stabilizing zero solution with respect to a part of the variables and is easily solved by applying the feedback linearization technique. The original problem is shown to reduce to a canonical form by applying a time-scale transformation and converting the intermediate affine system obtained to a normal form. It is noted that such a representation is not unique and depends on the choice of the time-scale transformation applied. The advantages and disadvantages of the three canonical representations obtained by means of three different, previously applied time-scale transformations are discussed. An example of the path following problem described by an affine system with a nonstationary drift field is presented.

The GNSS/INS Integrated System: Experimental Results and Its Application in Control of Mobile Robots

Two systems are described in this work. The first system is based on four dual-band GNSS (L1+L2, GPS/GLONASS) receiver boards. The second system consists of a single dual-band GNSS receiver and an INS block. Application of the integrated system in control of wheeled robots is considered. The control is the turning angle of the front wheels. The control goal is to bring the target point of the robot to a given trajectory and to stabilize its motion along that trajectory. The control synthesis problem is solved by applying the feedback linearization technique.