Creating Autonomous Multi-Object Safe Control via Different Forms of Neural Constraints of Dynamic Programming

Abstract

The aim of this work, which is an extension of previous research, is a comparative analysis of the results of the dynamic optimization of safe multi-object control, with different representations of the constraints of process state variables. These constraints are generated with an artificial neural network and take movable shapes in the form of a parabola, ellipse, hexagon, and circle. The developed algorithm allows one to determine a safe and optimal trajectory of an object when passing other multi-objects. The obtained results of the simulation tests of the algorithm allow for the selection of the best representation of the motion of passing objects in the form of neural constraints. Moreover, the obtained characteristics of the sensitivity of the object’s trajectory to the inaccuracy of the input data make it possible to select the best representation of the motion of other objects in the form of an excessive approximation area as neural constraints of the control process.

1. Introduction

Synthesizing the optimal control of single- and multi-object dynamic control processes is becoming an increasingly common challenge in practice. This applies to both non-autonomous and autonomous objects such as ships, cars, planes, and drones. One of many dynamic optimization methods can be used to solve this task.

Considering the significant impact of time-varying constraints on the control process, dynamic programming that meets Bellman’s optimality principle is the most effective optimization method, as presented in [1].

This research’s motivation was to expand the applications of the dynamic programming method to the synthesis of multi-object process control supported by artificial intelligence.

1.1. State of Knowledge

The scope of the literature presenting dynamic programming is wide, including software for solving dynamic programming tasks and applications in robotics, autonomous vehicles (AVs), unmanned air vehicles (UAVs), autonomous underwater vehicles (AUVs), and maritime autonomous surface ships (MASSs).

Regarding software for solving various practical dynamic programming tasks, we should first mention a tool called DynaProg developed by Miretti et al. and presented in [2]. It enables the optimization of deterministic, multi-stage, non-linear dynamic decision-making processes. In turn, in [3], Sundstrom et al. synthesized optimal control using dynamic programming in MATLAB version 2023 software. Knowledge of the control objective function and state equations of the control process with several state variables is required. Mikołajczak and Rumbut in [4] solved the problem of dynamic programming in a distributed artificial intelligence environment in C++ version 2023 software. Search management in a distributed development environment is optimized using Actors and Petri nets. Luus, in one of the chapters of the optimization encyclopedia [5], describes an iterative dynamic programming algorithm with variable stage lengths. An example of solving one of the very important optimization tasks using approximate dynamic programming is maximizing the number of survivors in a maritime disaster, as presented by Rempel et al. in [6], where the finite-horizon Markov decision process is used to describe the control process. Another category is the application of dynamic programming to solving the task of optimizing the course of a two-player game, which was presented by Gong et al. in [7]. The control objective function is the cost of capturing the target and has a linear–quadratic form.

Regarding the use of dynamic programming in robot design, the optimal control of tracking a robot manipulator with three degrees of freedom presented by Szuster and Gierlak in [8] using the dynamic programming method should be mentioned. The Lyapunov stability method is used to evaluate the optimization, and an artificial neural network is used to identify critical conditions. Then, the same authors in [9] presented the use of dual heuristic dynamic programming to solve the task of the optimal tracking control of an articulated robot manipulator. The duality of the control algorithm results from the two structures of the actor’s and the critic’s neural networks.

Many works have significantly contributed to the optimization of autonomous vehicle (AV) facilities. In [10], Kang and Jung presented the problem of identifying vehicle lanes using local line extraction and dynamic programming. The functional deviation of the actual vehicle movement line from the virtual straight line is minimized. In [11], Sundstrom et al. used dynamic programming to solve the problem of the optimal management of a hybrid vehicle’s electric energy. Thanks to this, a higher accuracy in achieving the global maximum was achieved. In [12], Silva and Sousa used dynamic programming to minimize a vehicle’s path deviation from a given path, considering uncertainties and environmental disturbances. In [13], Wei et al. presented the problem of optimizing the trajectories of many cars moving in a platoon formation, solved using dynamic and integer programming. In [14], Deshpande et al. solved the task of optimizing the energy consumption of connected and automated vehicles using dynamic programming. The process model was verified in real time. In [15], Lin et al. used dynamic programming to minimize the time needed for a group of vehicles to travel in a two-lane merging scenario. In [16], Wang et al. designed a real-time autonomous vehicle path-tracking controller using model-free adaptive dynamic programming. Compensation was made for non-linear external disturbances represented by an artificial neural network with a radial activation function. In [17], Lin et al. presented a programming algorithm for the dynamic planning and control of the path of an autonomous vehicle in situations of changing environments and surrounding vehicles.

Dynamic programming also has a large share in the optimization of unmanned air vehicles (UAVs). In [18], Gjorshevski et al. used a dynamic programming algorithm to determine the optimal trajectory in terms of the cost of the energy consumed by an unmanned aerial vehicle (UAV). Bouman et al. presented the use of dynamic programming to solve the traveling salesman problem with a drone in [19], using the example of optimizing product deliveries to customers using trucks and a drone. In [20], Flint and Fernandez solved the problem of planning the trajectories of many cooperating autonomous UAVs in the process of optimally searching an uncertain environment using dynamic programming. In [21], Azam et al. used a dynamic programming method to optimize the control of a swarm of UAVs, implementing the process of tracking the movement of multiple targets. Din et al. used dynamic programming in [22] to maximize the glide range performance of a UAV equipped with special actuating devices. In [23], Jennings et al. optimized the route planning of small unmanned aerial vehicles using dynamic programming, emphasizing the impact of wind disturbances.

Another area of application of dynamic programming is the optimization of various control tasks for autonomous underwater vehicles (AUVs). In [24], Vibhute used a dynamic programming method for the optimal motion control of an autonomous underwater vehicle (AUV), where the real-time identification of vehicle dynamics based on actuator input and obstacle position was used instead of a process model. In [25], Chen et al. used dynamic programming to optimally track the given trajectory of an autonomous underwater vehicle (AUV) using a fuzzy logic controller. In [26], Che and Hu solved the problem of the optimal tracking of an underactuated AUV’s trajectory using dynamic programming.

Dynamic programming is also used in synthesizing the optimal control of marine autonomous surface ships (MASSs). In [27], Wei and Zhou used dynamic programming to optimize the fuel consumption of a MASS along the weather route by controlling both the ship’s movement speed and heading changes. In [28], Geng et al. implemented an algorithm to plan the ship’s anti-collision route using dynamic programming using velocity obstacle models for stationary and moving targets. In [29], Esfahani and Szlapczynski used a hybrid method of dynamic programming and a learning system to stabilize the motion of an autonomous MASS under the influence of disturbances from wind, waves, and sea currents.

Among the most recent works, the paper by Mi et al. [30], in the field of safe predictive control based on a non-linear object model, is directly related to the topic of this article.

Since the optimization tasks using dynamic programming analyzed above do not concern safe and optimal multi-object control, this issue is the research goal of this article.

1.2. Study Objectives

The author’s main contribution is:

• Synthesis of safe multi-object control using dynamic programming;
• Experimental comparative analysis of a safe object trajectory with various forms of constraints to control the dynamic optimization process;
• Using the sensitivity characteristics of optimal control in assessing the form of process state constraints.

The novelty of the proposed solution to the task of safe control consists of:

• Mapping the motion of other encountered objects using neural constraints on the state of the control process;
• Development of an algorithm for determining the safe trajectory of an object in relation to a larger number of other objects encountered.

1.3. Article Content

First, the mathematical model of the multi-object control process is presented. Then, the synthesis of the algorithm for optimizing the object’s safe trajectory using the dynamic programming method is presented. The simulation studies of the developed algorithm present the safe and optimal object trajectories, as well as the characteristics of the optimization’s sensitivity to changes in the control process parameters. The conclusions include an analysis of the research results and the scope of further research.

2. Safe Multi-Object Control Process

The solution to the task of safe multi-object control consists of formulating an adequate model of the kinematics and dynamics of object 0 constituting the state equations, treating the motion of passing objects as kinematic constraints of the control process, and then using the dynamic programming method to determine the optimal and safe trajectory of object 0 (Figure 1).

2.1. Object 0 State Equations

The state equations of the object 0 motion control process are expressed in the following form:

$$\dot{x}=f\left(x, u,t\right)$$

(1)

where x is the state variable, u is the control, and t is the time, consisting of the kinematics and dynamics equations of its movement under the influence of rudder deflection α and changes in the propeller’s rotational speed n.

The kinematics of the movement of object 0 consists of the change in the time of its position coordinates p(X,Y):

$$\dot{X}=V_0\sin {\mathsf{\psi }}_0$$

(2)

$$\dot{Y}=V_0\cos {\mathsf{\psi }}_0$$

(3)

As an astatic control object in the ψ₀ course of anti-collision maneuvering using larger rudder deflections α, object 0 has the non-linear static characteristics ${\dot{\mathsf{\psi }}}_0=f\left(\mathsf{\alpha }\right)$, as shown in Figure 2.

The static characteristic ${\dot{\mathsf{\psi }}}_0=f\left(\mathsf{\alpha }\right)$ is identified during the spiral test of the object. Then, via regression analysis, the following mathematical description is obtained:

$$F\left({\dot{\mathsf{\psi }}}_0\right)=a_1{\dot{\mathsf{\psi }}}_0+a_2{{\dot{\mathsf{\psi }}}_0}^2+a_3{{\dot{\mathsf{\psi }}}_0}^3$$

(4)

Considering the above, the following second-order non-linear model of the object’s 0 dynamics is maintained:

$$T_1{\ddot{\mathsf{\psi }}}_0+F\left({\dot{\mathsf{\psi }}}_0\right)=k_1\mathsf{\alpha }$$

(5)

where T₁ is the time constant of the change in object 0’s course ψ₀ resulting from the rudder deflection α, and k₁ is the gain coefficient.

The dynamics of object 0 during the anti-collision maneuvering of its speed V₀ can be described with the following linear differential equation:

$$T_2{\dot{V}}_0+V_0=k_{2}n$$

(6)

where T₂ is the time constant, and k₂ is the gain coefficient.

To sum up, the general state Equation (1) of the object 0 control process takes the following form:

$${\dot{x}}_1=x_5\sin x_3$$

(7)

$${\dot{x}}_2=u_5\cos x_3$$

(8)

$${\dot{x}}_3=x_4$$

(9)

$${\dot{x}}_4=-\frac{a_1}{T_1}{x}_4-\frac{a_2}{T_1}{x}_4^2-\frac{a_3}{T_1}{x}_4^3+\frac{k_1}{T_1}{u}_1$$

(10)

$${ \dot{x}}_5=\frac{-1}{T_2}{x}_5+\frac{k_2}{T_2}{u}_2$$

(11)

$${\dot{x}}_6=1$$

(12)

where (x₁, x₂) are coordinates (X, Y) of object 0’s position; x₃ is object 0’s course ψ₀; x₄ is the angular velocity of object 0’s turn ${\dot{\mathsf{\psi }}}_0$; x₅ is object 0’s velocity; x₆ is the time; u₁ is the rudder angle; and u₂ is the rotational speed of propeller n.

2.2. Objects j Constraints

In the process model, the passed objects j constitute its constraints:

$$g_j\left(\mathit{x}, \mathit{u}, t\right)\ge 0$$

(13)

The following is related to the need to maintain a safe approach distance D_s:

$$D_{j,\mathrm{m}\mathrm{i}\mathrm{n}}\ge D_s$$

(14)

For a stationary object, the over-approach area will have the shape of a circle with a radius equal to the safe passing distance D_s under the given actual environmental conditions. For a moving object, we can choose the shape of the over-zoom area to consider the COLREG right-of-way rules. They separately apply to conditions of good visibility and limited visibility.

In good visibility conditions, object 0 must give way to object j on the right. However, if we encounter object j from the left, we expect it to give way to us. In conditions of limited visibility, the principle of giving way to the road on the right does not apply; only the increased value of the safe passing distance D_s applies. Then, the area of excessive approximation is assumed to be a circle with radius D_s.

The circular over-zoom area, as shown in Figure 3a, is described via the following relationship:

$$g_j^{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{l}\mathrm{e}}\left(X_j, Y_j\right)=X_j^2+Y_j^2-D_s^2\ge 0$$

(15)

where (X_j, Y_j) are the coordinates of the distance D_j of object j from object 0.

This article aimed to show how the shape of the area of excessive approach to object j affects the course of the determined safe and optimal trajectory of object 0.

Four forms of constraints g_j are proposed in conditions of good visibility: circular for objects j moving on the left; and hexagonal, elliptical, and parabolic for objects j moving on the right.

The dimensions of the hexagon, shown in Figure 3b, are determined by the dynamic length L_d and width W_d of object j:

$$L_{dj}=1.1L_j\left(1+0.345V_j^{1.6}\right)$$

(16)

$$W_{dj}=1.1\left(W_j+0.767L_j{V}_j^{0.4}\right)$$

(17)

where L_j and W_j are the length and width of object j, and V_j is the speed of object j.

The ellipse-shaped area, as shown in Figure 3c, is defined with the following inequality:

$$g_j^{\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{p}\mathrm{s}\mathrm{e}}\left(X_j, Y_j\right)={\left(X_j\sin {\mathsf{\psi }}_j+Y_j\cos {\mathsf{\psi }}_j-F_j\right)}^2{W}_{dj}^2+{\left(X_j\cos {\mathsf{\psi }}_j+Y_j\sin {\mathsf{\psi }}_j-F_j\right)}^2{L}_{dj}^2-L_{dj}^2{W}_{dj}^2\ge 0$$

(18)

The parabolic area of the excessive approximation of object j, as shown in Figure 3d, is described with the following relationship:

$$g_j^{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{l}\mathrm{a}}\left(X_j, Y_j\right)=X_j\sin {\mathsf{\psi }}_j+Y_j\cos {\mathsf{\psi }}_j-S{\left(X_j\cos {\mathsf{\psi }}_j+Y_j\sin {\mathsf{\psi }}_j\right)}^2\ge 0$$

(19)

where S is the span of the parabola arms.

3. Optimization of Safe Trajectory

The multi-object optimal control task formulated above consists of the state Equations (7)–(12) of object 0 containing the equations of its kinematics and dynamics, as well as the time-varying constraints (8) of the safe control process assigned to individual j objects. Many dynamic optimization methods can be used to solve this control problem, including Pontryagin’s maximum principle and Bellman’s optimality principle. The differences between these methods were described by Bokanowski et al. in [31].

Considering the multi-object control model formulated above, the most appropriate optimization method to use is the dynamic programming method based on the Bellman optimality principle.

3.1. Bellman’s Optimality Principle

The dynamics of the object are described with state Equation (1), which takes the following discrete form:

$${\mathit{x}}_{i+1}=\mathit{f}\left({\mathit{x}}_i, {\mathit{u}}_i\right)$$

(20)

which describes the evolution of the state $\mathit{x}\in {\mathbb{R}}^n$ from time i to i + 1, taking into account the control $\mathit{u}\in {\mathbb{R}}^m.$

The trajectory $\left\{\mathit{X}, \mathit{U}\right\}$ is a sequence of states X $\triangleq$ {x₀, x₁, …, x_N} and their corresponding controls $\mathit{U}$ $\triangleq$ {u₀, u₁, …, u_N−1} satisfying the state Equation (1). The total cost C of the control process is the sum of the current cost c_r and the final cost c_f incurred when moving the object from state x₀ and applying control $\mathit{U}$ until reaching the N horizon:

$$C\left({\mathit{x}}_0, \mathit{U}\right)={\sum }_{i=0}^{N-1}{c}_r\left({\mathit{x}}_i, {\mathit{u}}_i\right)+c_f\left(x_N\right)$$

(21)

The solution to the optimal control problem is to minimize the control sequence:

$${\mathit{U}}^{*}\triangleq \mathrm{a}\mathrm{r}\mathrm{g}\underset{\mathit{U}}{\min }C\left({\mathit{x}}_0, \mathit{U}\right)$$

(22)

Let $\mathit{U}$_i $\triangleq$ {u_i, u_i+1, …, u_N−1} be a control sequence. Then, the cost of implementing the control process C_i is the sum of the component costs from i to N:

$$C_i\left(\mathit{x}, {\mathit{U}}_i\right)={\sum }_{l=i}^{N-1}{c}_r\left({\mathit{x}}_l, {\mathit{u}}_l\right)+c_f\left(x_N\right)$$

(23)

The cost value at time i is the optimal implementation cost starting from x:

$$V_i\left(\mathit{x}\right)\triangleq \underset{{\mathit{U}}_i}{\min }{C}_i\left(\mathit{x}, {\mathit{U}}_i\right)$$

(24)

The end-time cost value is defined as $V_N\left(\mathit{x}\right)\triangleq c_f\left({\mathit{x}}_N\right)$.

At the final moment, the current cost is the implementation time of the optimal trajectory c_r = t_opt, and the final cost is the final deviation of the trajectory from the initial direction of movement of the object c_f = d.

Figure 4 shows the computational grid for the dynamic programming of the safe and minimum-time trajectory of object 0 according to the smallest path losses for safely passing the encountered j objects.

According to Bellman’s optimality principle, which is the basis of dynamic programming, optimal control from a given moment t to the final moment depends only on the current state of the process and control and does not depend on previous states and controls:

$$V\left(x, t\right)=\underset{u\left(t\right)}{\min }\left\{c_r\left[\left(x\left(t\right), u\left(t\right)\right)\right]+V\left[f\left(x, u, t-1\right)\right]\right\}$$

(25)

taking into account the constraints (8) of the control process state.

3.2. Algorithm

To determine the optimal trajectory of object 0 in situations of safely passing encountered j objects, an appropriate control algorithm was synthesized in MATLAB 2023 version software, the functional diagram of which is presented in Figure 5.

The state plane is divided along the path of object 0 into N stages, and each stage is divided into nodes. At each node, the movement time from the previous node is calculated.

At the same time, according to Algorithm 1, areas of excessive proximity are determined at the time of stay in a given node.

Algorithm 1: Hexagon, ellipse, parabola, and circle domains calculation

Begin 1. For n = 1; 2. For Neural Network Calculation; 3. For Visibility; 4. If Good; Calculation dimensions hexagon or ellipse or parabola; Else Calculation dimensions circle; End 5. Domains plotting; End End 6. For Domains crossing; 7. If Yes; Speed = 0; Else Speed = V; End End Saving data; 8. If n = N; else n = n + 1; End End

If a given node is within the bounding box of object j, a penalty of 10⁶ s is taken as the movement time. Then, the calculation of the object’s displacement time is repeated for each subsequent node and possible path between nodes. Before moving on to the next node, the smallest shift time value and node number are selected and stored. In the last stage, as many possible paths will be determined as there are nodes at one stage, from which the path with the smallest time to move the object from the starting node to the final node is selected. By connecting subsequent nodes from end to beginning, an optimal safe trajectory of the object is obtained.

However, the areas of excessive zoom in the form of a circle, hexagon, ellipse, and parabola do not have a constant size but increase as object j approaches object 0. An artificial neural network was used for this purpose, the output of which constitutes the risk of collision with a given object, as shown in Figure 6 [32].

The neural network, implemented in MATLAB 2023 version software in the Neural Network Toolbox, consists of three layers of neurons, with non-linear activation functions in the first and second layers and a sigmoid activation function in the third output layer. To learn it via experienced navigators, an error propagation algorithm with an adaptive learning rate and momentum was used.

In the last stage N of calculations, the path with the smallest object 0 travel time t_opt is selected, and the entire trajectory to the beginning is determined. In this way, a safe and optimal trajectory of object 0 is obtained when passing encountered j objects.

The dynamic programming method discussed in this article differs in the generation of time-varying constraints on the state of the control process by an artificial neural network from other similar applications, for example, those presented by Shi et al. in [33,34] regarding the Q-learning algorithm for a batch tracking fault-tolerant process.

4. Computer Simulation

Simulation studies of safe object control algorithms in collision situations were conducted using the Automatic Radar Plotting Aid (ARPA) laboratory simulator software or in Matlab/Simulink 2024a software based on real navigation situations previously recorded on the object.

The evaluation of the developed algorithm for determining a safe and optimal trajectory was carried out via a computer simulation of the example of a navigation situation of passing object 0 with j = 6 encountered objects (

Table 1. Values, measured in the ARPA anti-collision system, of the state variables of the process of controlling the movement of object 0 and j = 6 passing objects.

Object j	Bearing Nj (deg)	Distance Dj (nm)	Course ψj (deg)	Speed Vj (kn)
0	0	0	0	20
1	11	7.5	200	16
2	305	4	0	0
3	55	5.7	90	12
4	326	8.8	90	14.5
5	340	12	0	0
6	6	14.3	180	16.2

).

First, the courses of safe and optimal trajectories of object 0 in good visibility conditions were calculated for various forms of excessive approach areas: hexagonal and circular (Figure 7a), parabolic and circular (Figure 7b), elliptical and circular (Figure 7c), and only circular (Figure 7d).

Comparing the safe and optimal object trajectories for various forms of closest approach areas, it can be concluded that the lowest value of the optimization criterion is provided by circular areas, and the smallest by hexagonal areas.

Then, safe and optimal trajectories of object 0 were determined in conditions of limited visibility when maneuvering by only changing the course of object 0 (Figure 8a), and when maneuvering by changing the course and speed (Figure 8b).

Another simulation of the algorithm was performed for a larger number of encountered objects j to show an important advantage of the dynamic programming of a safe and optimal trajectory of object 0. Namely, the more constraints assigned to each object j, the shorter the trajectory calculation time.

Figure 9a shows the safe and optimal trajectory of object 0 in conditions of good visibility, and Figure 9b in conditions of limited visibility.

An important test for evaluating dynamic programming algorithms is the examination of their sensitivity to changes in various parameters of the control process [35,36,37].

This paper examines the sensitivity of the optimal trajectory both to the inaccuracy of the input data and changes in environmental conditions, and to the density of the dynamic programming mesh.

The measure of sensitivity s was the relative change in the optimal time t_opt of the safe and optimal trajectory of object 0 caused by a change in the selected parameter of the control process p:

$$s=\left|\frac{t_{opt}\left(p\right)-t_{opt}\left(p\pm \delta p\right)}{t_{opt}\left(p\right)}\right|100\%$$

(26)

As a result of repeated simulations of the algorithm for determining the safe trajectory of object 0, the sensitivity characteristics s of the t_opt criterion optimization to changes in the following parameters were obtained: the velocity δV and course δψ of object 0 (Figure 10), the velocity δV_j and course δψ_j of object j (Figure 11), the distance δD_j and bearing δN_j to object j (Figure 12), the distance D_s of safe passing of objects 0 and j, and the mesh density Δx of dynamic programming (Figure 13).

Due to the inaccuracy of the measurements of the speed V and course ψ of object 0, the greatest sensitivity s of the optimization occurs for hexagonal areas and the lowest for parabolic areas of the closest approximation of objects 0 and j.

Similarly, with the inaccuracy of the measurements of the speed Vj and course ψ_j of object j, the greatest sensitivity of the optimization occurs for hexagonal areas and the lowest for parabolic areas of the closest approximation of objects 0 and j.

However, with the inaccuracy of the measurements of the distance D_j and bearing N_j to object j, the greatest sensitivity of optimization occurs for ellipsoidal areas and the lowest for parabolic areas of the closest approach of objects 0 and j.

When increasing the value of the safe passing distance D_s, circular areas of the closest proximity of objects 0 and j cause the greatest optimization sensitivity, and hexagonal areas of the closest proximity to objects 0 and j cause the least.

However, when reducing the value of the safe passing distance D_s, the hexagonal areas of the greatest proximity of objects 0 and j cause the greatest optimization sensitivity, while the parabolic areas of the closest proximity of objects 0 and j cause the least.

5. Conclusions

The synthesis of the optimization of the safe and optimal trajectory of an object when passing other encountered objects according to the Bellman optimality principle, and then its presentation as an appropriate control algorithm and its simulation studies allow the following conclusions to be formulated:

• It is possible to formulate an adequate model of the actual multi-object control process, which allows for its optimization while maintaining safe traffic conditions;
• Optimization using the Bellman optimality principle in a non-linear model of the control process correctly reproduces the kinematics and dynamics of moving objects;
• The limitations of the optimality principle allow for an adequate representation of the actual control process;
• By formulating various forms of limitations in the over-approach areas, it is possible to select their best form ensuring the least sensitivity to changes in the parameters of the control process;
• The optimality principle algorithm allows us to consider a larger number of objects whose data come from ARPA anti-collision radar systems;
• The stability of safe control is ensured in the real facility by the heading change PID controller with selected optimal settings, for example, according to the Ziegler–Nichols stability criterion.
• The effect of the developed method of optimizing the object’s trajectory is the current mapping of the collision risk through the size of the neural areas of prohibited maneuvers;
• The advantage of the dynamic programming algorithm compared to the second basic optimization method is that the maximum principle is its computational efficiency in real time of the control process; thus, the more state constraints assigned to each encountered object, the shorter the time to determine its safe trajectory.

Future research on the presented topic may include the following:

• Other forms of process state constraints;
• Adapting the final conditions of the trajectory optimization task to the course of the actual control process;
• A comparison of the results of object trajectory optimization according to the optimality principle with optimization results obtained using other static, dynamic, and game optimization methods.