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The paper is devoted to several examples of control algorithm development for two-wheeled double-track robot and low-tonnage vessel-catamaran with two Azipods that show practical aspects of the application of one nonlinear system control method — terminal state method. This method, developed by the authors of the present paper, belongs to the class of methods for inverse dynamics problem solving. Mathematical models of control objects in the form of normal systems of third-order nonlinear differential equations for the wheeled robot and seventh-order ones for the vessel are presented. Design formulas of the method in general form for terminal and stabilizing controls are shown. A routine of obtaining calculation expressions for control actions is shown. Results of computer simulation of bringing the robot to a given point in a given time, as well as bringing the vessel to a given course during a “strong” maneuver, are described.

Currently, ultimate nonlinear system modes often arise in control problems. In such cases, classical methods based on the linear automatic control theory are unusable. Due to it, more and more attention is being paid to the development and improvement of nonlinear system control methods. We can single out several approaches that gave rise to most nonlinear system control methods. They include methods of solving the inverse mechanics (dynamics) problem [

The paper is devoted to the questions of practical application of the method under discussion in cases when TSM-control calculations are rather simple. So are the terminal problem of bringing a two-wheeled double-track mobile robot to a given point and the problem of bringing the vessel at a given heading angle. Herewith, it is required to ensure the possibility of pursuing the so-called “strong” vessel maneuvers. These are, for instance, maneuvers with a large angle of deviation of the steering components (in this paper, Azipod).

Two-wheeled two-track robots are employed as amphibious platforms for observation and measurement. For example, a similar robot with crawler is proposed in [

A mathematical model of a two-wheeled double-track robot is given in paper [

The second problem being solved is a part of a larger problem, which is to develop a crew-free operation mode for a small-sized robotic research vessel designed for environmental monitoring of the marine coastal area. In this case, a version of TSM with stabilizing (tracking) type criterion was used. To keep the vessel on a previously specified route, it is necessary to control it in automatic mode so that in case of deviation from the route the vessel returns to the route. In this case, there may be situations (sudden obstacle avoidance, moving in narrow areas, etc.) when the vessel maneuver is performed at a large rudder angle or Azimuthing Electric Propulsion Drive (Azipod) rotation. Then vessel dynamics becomes nonlinear and therefore the appropriate control methods should be applied. In navigation such vessel maneuvers are called “strong”.

The specifics of the problem to keep the vessel on a route and the vessel mathematical model limit the range of possible control methods appropriate for obtaining a solution. In particular, it refers to the nonlinear vessel model we have obtained [

For bow maneuvering devices, it is proposed to use additionally the algorithm in the form of differential equation relative to the transverse control force [

Herewith algorithms of calculating anticipatory points are used. From these points precalculated program controls are pursued to turn the vessel to a new course. Conditions of the stabilization along a given course

A common drawback of these methods, as well as others based on the use of linear feedback from the state coordinates [

Often during the nonlinear system synthesis, methods of solving the inverse dynamics problem are used. These include, in particular, the input-output linearization method [

Terminal state method was developed to solve terminal control problems for nonlinear affine systems in the form of (

For differential systems with smooth constraints, criterion problem is formulated according to (

According to this method, control (TSM control) is calculated by the following expressions:

Obtained controls may be constrained by amplitude limitations, if necessary. Such constraints in the active state affect deviation from the given (e.g., exponential) form of the criterion function. At the same time, control goal—to achieve a given terminal state—is commonly accomplished. Sometimes, for the process of control to be sustainable, artificial imposition of restrictions is preferable.

In (

It is important to realize that for calculations in (

For stabilization problems, the algorithm formed from (

Then the criterion transforms to

The robot model excluding the wheel drives dynamics description, i.e., its kinematic model, has the following form [

The kinematic model defined by (

First of all, it should be noted that models of this type take place in the case of using stepper driving motors, as well as when using crawler propulsions.

Some features of this model are as follows: the right sides of its differential equations are independent from the state vector; controls

Mathematical model was obtained in [

Mathematical model has the form of (

Herewith,

State functions

In (

Masses and moment of inertia from (

To obtain control formula considering Azipod drives dynamics instead of (

Equating right sides of (

A. Isidori’s linearizing by input/output method [

(

In (

For the criterion function, the following quadratic form is used:

The sense of criterion by (

The output of the controls evaluated by (

System of (

In [

The same algorithm is implemented in the onboard C++ version of a real robot. As follows from the kinematic scheme, which is the basis for (

Consequently, equations (

This is clear from (

The resulting system is

From (

However, it is more interesting to bring the robot to a given point

For this purpose, the control problem should be modified by assigning the other criterion function instead of (

This function reflects the following control objective: bringing the coordinate

The latter are now much more complicated than in the case of criterion in the form of (

Control by (

Time dependence of the robot left wheel coordinate.

Time dependence of the robot right wheel coordinate.

Robot trajectory.

Figures

Time dependence of the robot left wheel angular velocity (control action).

Time dependence of the robot right wheel angular velocity (control action).

For the denoted problem, criterion by (

It should be noted that we had to add a term containing

The matrices from the system of (

For the case when the drive (integrating) inertia is not taken into account, we have

Herewith, according to (

The expression for the pseudoinverse matrix is the following:

and the expression for TSV is

As a result, we obtain the following expression for the control action at the integrating drive input:

In the algorithm of (

Control by (

In contrast to the problem solved in the previous paragraph, now let us set the problem of controlling the vessel course by means of an implicit criterion:

The second expression of (

To evaluate the control, we apply (

In accordance with (

Then, for

Simulation showed that, in any case, for the vessel in question, instead of the latter formula, the simplified one can be used. This follows from the fact that

In (

Control by (

Thus, two algorithms for bringing the vessel to a given course were obtained: by means of control by (

To achieve both simplicity and accuracy of computations, a composite algorithm is offered. This algorithm includes a “strong” maneuver through the control by (

Applying composite algorithm makes sense in the case of using onboard controllers, intended to perform simple computational operations. This is of course not typical for the catamaran vessel in question. It is more relevant to ordinary small vessels intended for everyday purposes, for example, for cleaning leaves and relatively large debris from the water surface.

Simulation results in the form of the time

It is possible to regulate both trajectory and time of bringing the vessel to the specified course by choosing parameters

Figures

Vessel trajectory during rotation towards the return course for

Time dependence of the vessel heading angle during rotation towards the return course for

Figures

Vessel trajectory during slowed rotation at a given heading angle of 57.3 degrees for values

Time dependence of the vessel heading angle during slowed rotation at a given heading angle of 57.3 degrees for values

A certain influence on the trajectory is also provided by the value of the free parameter

Graphs shown in Figures

Vessel trajectory during accelerated rotation at a given heading angle of 57.3 degrees for values

Time dependence of the vessel heading angle during accelerated rotation at a given heading angle of 57.3 degrees for values

Vessel trajectory during rotation towards the return course for values

Time dependence of the heading angle during rotation towards the return course for values

We note that in all the cases presented, the control error is close to zero (in the simulation, the first significant digit is in fourth place after the decimal point). By control error, we mean the difference between specified and attained values of the course.

Graphs in Figures

Vessel trajectory during rotation towards the return course for values

Time dependence of the heading angle during rotation towards the return course for values

Simulation was performed for two consecutive time intervals with a duration of

As can be seen from the graphs of Figures

It is important to note that the control by (

It should be noted that the results concerning vessel control adduced in this paper represent development of methods and algorithms published in [

The practical aspects of using the author’s nonlinear systems control method—terminal state method (TSM)—are discussed in the paper. Expressions for control calculating according to TSM with respect to terminal and stabilization (tracking) problems are presented. A special feature of the method is the use of terminal state variables (TSV), whose sense is the forecast of uncontrolled system final state, dependent on the current time and current state. Two examples of the method application are provided. The first one is a terminal state control of a two-wheeled double-track mobile robot. The second one is a control of the course of a small-capacity vessel-catamaran with Azipods during the “strong” maneuvers performing, i.e., maneuvers at large rudder angles. Two TSM algorithms are proposed for the course control. The first one is based on a quadratic criterion for deviations from a given course. The second one is based on the criterion for tracking the vessel angular velocity in relation to its center of mass. The first algorithm is proven to be simpler from a computational point of view, but gives a noticeable static error. The second algorithm is more complicated but has a zero static error. A composite algorithm, consisting of a consecutive implementation of the first and second algorithms, combines the advantages of both algorithms. Mathematical models of two control objects and expressions for control are presented. A routine of obtaining expressions for control actions is described. Results of computer simulation in the form of graphs of time dependencies for robot and vessel current coordinates, their trajectories, and robot control actions are shown.

The Mathcad program data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The studies were pursued with the state financial support of the Ministry of education and science of Russia (unique project ID RFMEFI57817X0259).