Подход к отслеживанию траектории многороторных летательных аппаратов в неизвестных условиях / Trajectory Tracking Approach for Multi-rotor Aerial Vehicles in Unknown Environments тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Кулатхунга Мудийанселаге Гисара Пратхап Кулатхунга

• Кулатхунга Мудийанселаге Гисара Пратхап Кулатхунга
• кандидат наук
• 2024

Оглавление диссертации кандидат наук Кулатхунга Мудийанселаге Гисара Пратхап Кулатхунга

CONTENT

INTRODUCTION

CHAPTER 1. LITERATURE REVIEW

1.1 Introduction

1.2 Motion model selection

1.2.1 Exact model

1.2.2 Empirical model

1.2.3 Differential fatness

1.3 Initial waypoints identification

1.4 Initial trajectory generation

1.4.1 Define trajectory

1.4.2 Minimum-snap based trajectory generation

1.4.3 Polynomial trajectory generation as QP

1.4.4 Unconstrained polynomial trajectory generation

1.4.5 Unconstrained polynomial trajectory generation with

collision avoidance

1.4.6 Covariant gradients for trajectory generation

1.4.7 B-spline based trajectory generation

1.4.8 Bernstein piecewise trajectory generation

1.4.9 Comparison of several trajectory techniques

1.5 Free space extraction

1.6 Continuous trajectory refinement

1.7 Receding horizon trajectory planning

1.7.1 LQR-based trajectory generation

1.7.2 MPC-based trajectory generation

1.7.3 Disturbance estimation

1.8 Solving trajectory planning problem

1.9 Summary

CHAPTER 2. THE HARD CONSTRAINTS-BASED PLANNER

2.1 Introduction

2.2 Related work

2.2.1 Map building and motion model selection

2.2.2 Trajectory generation

2.2.3 Trajectory tracking

2.2.4 Trajectory generation and tracking with model predictive control

2.2.5 Trajectory planning with Gaussian process

2.3 Appropriate motion model selection

2.4 Environment reasoning for search space construction

2.5 Reference trajectory generation

2.6 Trajectory tracking problem formulation without considering search space constraints

2.7 Trajectory tracking problem formulation considering search space constraints

2.7.1 Formulation with multiple-shooting

2.7.2 Formulation with direct-collocation

2.8 Data-driven residual dynamics learning

2.8.1 Residual augmented quadrotor motion model

2.8.2 Augmented residual dynamics learning with GP

2.8.3 Sparse GP Regression

2.8.4 Tracking problem synthesis using a residual augmented

motion model

2.9 Summary

CHAPTER 3. THE SOFT CONSTRAINTS-BASED PLANNER

3.1 Introduction

3.2 Related work

3.3 The soft constraints-based planner problem formulation

3.3.1 Cost for search space constraints

3.3.2 Finding pushing direction

3.3.3 Parallel convex decomposition

3.3.4 Calculating gradients

3.3.5 Cost of smoothing

3.3.6 Cost of feasibility

3.4 Dead zone recovery

3.5 Summary

CHAPTER 4. THE CASCADE PLANNER (HYBRID

APPROACH)

4.1 Introduction

4.1.1 Characteristics and functions of a hard constraints-based planner

4.1.2 Characteristics and functions of a soft constraints-based planner

4.2 The main components of the proposed cascade planner

4.2.1 Soft constraints-based planner

4.2.2 Hard constraints-based planner

4.2.3 Fail-safe recovery mechanism

4.3 The complete framework

4.3.1 Framework state transition

4.3.2 Hyper-parameters configuration

4.4 Analysis of residual dynamic learning

4.4.1 Latent distribution of residual dynamics approximation

4.4.2 Computation footprint vs residual dynamics estimation accuracy

4.4.3 Assessing the effect introducing residual dynamics produces

on the nominal dynamics

4.5 Analysis of the SLQG-based design effect of smoothing the control policy generation

4.6 Average-runtime estimation in real-world condition

4.7 Summary

CHAPTER 5. EXPERIMENTAL SETUP AND RESULTS

5.1 Experimental setup

5.2 The accuracy of the reference trajectory tracking without considering search space constraints in simulated vs real-world conditions

5.3 Reference trajectory tracking accuracy vs the maximum allowed

speed for MAV

5.4 Reference trajectory tracking accuracy of the hard constraints-based planner

5.5 Reference trajectory tracking accuracy using the proposed dataset

5.6 The behaviour of the proposed approach in a real-world condition

5.7 Comparative analysis of the proposed approach with nominal

system model

5.8 Comparative analysis of the proposed approach with augmented residual model

5.9 Summary

CONCLUSIONS AND PERSPECTIVES

LIST OF ABBREVIATIONS

LIST OF NOTATIONS

LIST OF GRAPHIC MATERIALS

LIST OF FIGURES

LIST OF TABLES

Bibliography

Введение диссертации (часть автореферата) на тему «Подход к отслеживанию траектории многороторных летательных аппаратов в неизвестных условиях / Trajectory Tracking Approach for Multi-rotor Aerial Vehicles in Unknown Environments»

INTRODUCTION

In recent years, Multirotor Aerial Vehicles (MAVs)-related manifestations, e.g., trajectory planning, path following, and exploration, expanding the range for venturing out computationally expensive techniques in various disciplines, including agriculture, aerial photography, and crop monitoring. Such advancement is mainly due to modern methods of mathematical modelling that significantly boost the performance of the algorithm execution time. Furthermore, the recent development of numerical optimization tools that can run on lightweight embedded devices, the recent progression in computation capabilities, and the boost of embedded sensing capabilities have also helped the advancement of MAVs-related manifestations. There are various challenges to be addressed when working with trajectory planning-related problems. One such pivotal problem is the reference trajectory tracking problem that is the focus of the thesis, which can be applied in many aforementioned manifestations. Reference Trajectory tracking requires tracking a specified trajectory at each given time, where trajectory can be considered as a dth order time-parameterized polynomial that must ensure continuity and smoothness. The general idea of reference trajectory tracking problem formulation is shown in Fig. 0.1.

Figure 0.1 — The general idea of reference trajectory problem formulation.

The thesis objective concentrates on reference trajectory tracking in the listed disciplines. In such disciplines, search space construction is a rather complicated task due to the dynamic search space constraints. Moreover, the free space is entirely or partially unknown. Furthermore, unpredictable events can occur at any time due to numerous reasons. Thus, to tackle those unexpected problems in real time, a fast and accurate constrained optimization technique is required. In general, such a constrained optimization problem is divided into a few subcategories: path planning followed by smoothing, kinodynamic search-based trajectory generation, cascade planning, and motion primitive-based approaches. Among them, cascade planning approaches are the most widely used and efficient way to address the considered problem compared to the other approaches. In this regard, it is necessary to develop a new cascade planning method (a cascade mathematical model) that takes into account search space constraints as part of the provided constraints that minimize the designed cost function.

The goal

The goal of the thesis is to develop a cascade mathematical model for reference trajectory tracking as a constrained optimization problem. The proposed cascade mathematical model comprises two sub-models: a soft constraints-based nonlinear optimization problem (soft constraints-based planner) and a hard constrained-based nonlinear optimization problem (hard constraints-based planner), both work in parallel. The soft constraints-based planner refines the initial reference trajectory iteratively when the trajectory goes within the search space constraints and lets the hard constraints-based planner optimizes its objective ensuring the provided constraints at every planning step by solving sequential quadratic program.

Research objectives

The thesis goal was achieved by addressing the following research objectives:

— Reduce computational footprint - Use well-known mathematical modelling technique, Stochastic Linear Quadratic Gaussian, to model the hard constraints-based planner for achieving long-range trajectory tracking ensuring the provided constraints: search space constraints

— Ensure provided constraints while avoiding trapping in local minima - Propose a cascade mathematical method: a hard constraints-based planner consider search space constraints and a soft constraints-based planner avoids trapping in local minima.

— Improve the smoothness of trajectory adhering to the provided constraints - Model the cost function that minimizes the effects from higher-order components: second-order derivative (acceleration) and third-order derivative (jerk), that reduces residual dynamics between the proposed cascade mathematical model and actual robot model.

To achieve these listed objectives, several modern mathematical modelling tools were used: CasADi is for nonlinear optimization and algorithmic differentiation, MOSEK is for modelling and solving convex optimization problems, and LBFGS++ is for unconstrained minimization problems.

Main results of the dissertation

The main results of the dissertation are as follows:

— A Stochastic Linear-Quadratic Gaussian(SLQG)-based model to overcome computational demands with a short prediction horizon to obtain real-time performance.

— A Sparse Gaussian Process(SGP)-based model to alleviate the model residual between the approximated mathematical model and actual robot model.

— A soft constrained-optimization technique to refine the reference trajectory by pushing towards the free space in the global phase.

— A new dataset was published for benchmarking purposes

Scientific novelty

The research work presents several novel scientific results that come within 1.2.2. «Mathematical modeling, numerical methods and complexes of programs» speciality. Mathematical modelling was applied at the design stages of reference trajectory tracking problem formulation. Such modelling helped to obtain real-time performances ensuring the provided consideration. The obtained novel scientific results are listed as follows:

— A mathematical modelling approach for trajectory refinement: a soft constrained-optimization problem that continuously refines the reference trajectory towards the free space in the global phase minimizing its cost.

— Two mathematical modelling approaches that are based on convex programming: first, helps to find the direction of the gradients that the reference trajectory must be pushed out, and second, recover the reference trajectory if the first approach fails to find the desired gradients.

— A mathematical modelling approach based on Stochastic Linear-Quadratic Gaussian-(SLQG): a hard constrained-optimization problem with an approximated mathematical model to overcome computational demands with a short prediction horizon to obtain real-time performance.

— A mathematical modelling mechanism for residual dynamics learning: a Sparse Gaussian Process(SGP)-based model to alleviate the model residual between the approximated mathematical model and actual robot model.

— A software package that comprises all the listed approaches is for performing computational experiments. The software package was numerically verified in a simulated environment and tested in real hardware.

In general, the proposed cascade mathematical model numerically solves two trajectories in parallel, the first one, a soft constraints-based planner, tries to refine the initial reference trajectory pushing the reference trajectory towards the free space, while the second one, a hard constraints-based planner, optimizes its objective ensuring the provided constraints at every planning step incorporating the refined trajectory. Since the first pushes the reference trajectory towards the free space, the number of constraints applied to the second is less. Such cascade mathematical

modelling helps improve the average run-time of the hard constraints-based planner as well as the overall performance of the proposed cascade mathematical model.

Compliance with the specialty passport

The subject of study is to develop a cascade mathematical model for

reference trajectory tracking as a constrained optimization problem.

The field of study corresponds to the following points of the passport of speciality 1.2.2. - «Mathematical modeling, numerical methods and complexes of programs» (Technical Sciences) in the following points:

— Point 2. Development of qualitative and approximate analytical methods for the study of mathematical models;

— Point 3. Development, justification and testing of effective computational methods using modern computer technologies;

— Point 4. Implementation of effective numerical methods and algorithms in the form of complexes of problem-oriented programs for conducting a computational experiment;

— Point 5. Development of new mathematical methods and algorithms for verifying the adequacy of mathematical models of objects based on data from a field experiment;

— Point 6. Development of computer and simulation systems, algorithms and simulation methods based on analysis.

Theoretical and practical significance

A complete software framework that focuses on reference trajectory tracking was modelled as a constrained-optimization problem and developed using modern mathematical modelling tools (CasADi, MOSEK, and LBFGS++).

Theoretical significance The purpose of the dissertation is to improve mathematical methods and algorithms for modelling of reference trajectory tracking

problem formulation as a constrained-optimization problem. Thus, the following theoretical research problems have been addressed to achieve the thesis objectives:

— How to reduce the computational footprint

— How to improve the smoothness of refined reference trajectory

— How to ensure hard constraints and avoid trapping in local minima

— How to obtain real-time performance

To address the listed research problems, initially, the reference trajectory tracking problem was formulated without considering search space constraints. In that tracking problem formulation, a linearized reference trajectory error model was considered. A single shooting technique, Linear Quadratic Regulator (LQR), was proposed to minimize reference trajectory tracking errors. However, in the thesis, it is shown that LQR is not appropriate to handle nonlinear constraints, e.g., static and dynamic search space constraints. Hence, Nonlinear Model Predictive Control (NMPC)-based technique was proposed to handle such nonlinear constraints while keeping the error model without linearizing. Afterwards, NMPC was formulated using direct collocation and multiple shooting. However, multiple shooting was identified as the correct way to solve NMPC when considering computational constraints and accuracy. Since the onboard computer has low computational power, such constraints, that are imposed when formulating the NMPC have to be minimized. Hence, a soft constraints-based planner was proposed to iteratively refine the original reference trajectory, pushing towards the free space, and thus reducing the number of search space constraints applied to the NMPC. The algorithm complexity of the soft constraints-based planner was proved using Big O notation. Since the number of static and dynamic search space constraints is subject to change in each time instance of planning, global optimality can not be guaranteed by the hard constraints-based planner, NMPC formulation. Since NMPC is a heuristic approach, the feasibility of the trajectory is guaranteed ensuring the provided constraints.

Practical significance lies in the development of the proposed framework and the experimental procedures that were carried out in various simulated and real-world experiments, e.g., static and dynamic search space constraints

were present in both real-world and simulated experiments. Since the proposed framework concentrates on applications such as crop monitoring in agriculture, aerial photography in geodesy and construction, and mapping in an agricultural field. Such a profile can be modelled as a set of hard constraints or box constraints into the proposed cascade mathematical model, the practical relevance is quite high.

Research methodology and methods

To achieve the thesis goals and objectives, modern mathematical modelling-based techniques were applied. The obtained results, propositions and conclusions are based on the methods of computational geometry, the theory of simulation modelling, the theory of differential equations, the theory of variation of calculus, the theory of experimental planning, and the theory of mathematical statistics. To implement the set of goals and objectives, the sparse matrix library, namely, Eigen, was used in C++11. The ROS software stack was used to implement the complete framework. To verify the obtained theoretical results, Gazebo was used with a PX4-enabled quadrotor computer simulation engine. DJI M100 quadrotor was used for real-world experiments, where the Gazebo PX4-enabled quadrotor node was replaced with the DJI M100 node for real-world experiments.

Accuracy of obtained results

The proposed cascade mathematical model (or the cascade planner), consists of two sub-models: a hard constraints-based planner and a soft constraints-based planner. The hard constraints-based planner was designed using approximated nominal motion model. Later, an augmented residual dynamics learning model was proposed to reduce the residual dynamics between the cascade planner and the actual robot model. The hard constraints-based planner optimizes its objective ensuring the provided constraints without the intervention of the soft constraints-based planner. The accuracy (or reference trajectory tracking error) of the cascade planner was estimated considering all the listed characteristics of the proposed cascade planner.

The first experiment was to estimate the accuracy of reference trajectory tracking without considering search space constraints in simulated vs real-world conditions. In the simulated conditions, the average velocity error was 0.02± 0.01m/s, and in the real-world condition, the average velocity error was 0.07 ± 0.02m/s, for tracking a provided reference trajectory. The next experiment was devoted to estimating the accuracy of reference trajectory tracking by incrementing the max allowed speed. The tracking accuracy decreases when increasing the MAV's velocity because of the approximated nominal motion model, i.e., the model can not handle dynamic changes. Afterwards, the trajectory tracking accuracy of the hard constraints-based planner with two other approaches: presented by [1] and by [2], were compared with the proposed approach. The same validation technique, which [1] used, was applied to check the behaviour when hard constraints are varied. Ten Poisson forests [3] with 10x10x10 of densities in between 0.2 trees/m and 0.8 trees/m were generated. The experiment result showed that the proposed approach has a high success rate in terms of reaching the goal. However, the average run-time (0.1 ± 0.05 s) was considerably higher compared to the others.

The last two experiments were devoted to a comparative analysis of the proposed approach with nominal and augmented residual models. In the first experiment, 12 random forests with a density of (40mx40mx 10m) were generated. The three other methods that were used are RRT* [4], a hard constraints-based planner [5], and FASTER [6], to validate the proposed cascade planner. The success rate was higher compared to the others. Moreover, the average run-time (0.03±0.01 s) has improved after introducing the soft constraints-based planner along with the hard constraints-based planner. In the second experiment, the same experimental setup was used as in the first experiment. However, four different trajectory planners: sampling-based grid search followed by trajectory generation, NMPC-based replanner [4], hybrid planner (local and global) [7], and KinoJGM [8] were validated against the proposed. All the selected planners have their advantages and limitations concerning navigation speed it can fly. Even after incorporating residual dynamics learning, the average run-time remains around 0.4 ± 0.01s and has the highest success rate compared to the other four approaches.

Assertions that are presented for defense

— A Stochastic Linear-Quadratic Gaussian(SLQG)-based model to overcome computational demands with a short prediction horizon to obtain real-time performance.

— A Sparse Gaussian Process(SGP)-based model to alleviate the model residual between the approximated mathematical model and actual robot model.

— A soft constrained-optimization technique to refine the reference trajectory by pushing towards the free space in the global phase.

— A new dataset was published for benchmarking purposes.

The reliability of the results obtained in the dissertation research is confirmed by:

— Time-complexity analysis for proving the soundness of the proposed approach;

— Numerical results of real-world and simulated experiments that showcase the agile flights in various unknown cluttered environments;

— Published scientific papers in peer-reviewed publications, including at flagship international conferences and top journals (two Q1 journals);

— Releasing a new dataset that was used for benchmarking our approach with the three other approaches, the dataset consists of a hundred trajectories across cluttered environments. This dataset can be used for evaluating the performance of reference trajectory tracking approaches.

Approbation of research results

The main findings of the dissertation are detailed in six printed publications. Four of these have been published in international journals, all of which are Q1 journals: Nature Scientific Reports, IEEE Robotics and Automation Letters, the Journal of Field Robotics, and Remote Sensing. The remaining two publications are in journals listed by the Russian Higher Attestation Commission. Also, the main results were reported at international conferences: International Conference on Robotics and Automation (ICRA) and Artificial Intelligence Journey. Five of the journals are indexed in RSCI and WoS.

Personal contribution of the author

The author independently analyzed the literature on the thesis topic, defined the goals, objectives, choice and use of quadrotor, sensors, mathematical modelling methods, numerical methods, and software packages results of the dissertation research submitted for defence and included the main statements, approbation, and personally published the results in scientific articles under the scientific supervision of Dr Alexandr Klimchik.

Заключение диссертации по теме «Другие cпециальности», Кулатхунга Мудийанселаге Гисара Пратхап Кулатхунга

CONCLUSIONS AND PERSPECTIVES

Conclusion of the thesis

This thesis is devoted to developing a reference trajectory tracking approach considering dynamic uncertainty in unknown environments whilst ensuring dynamic feasibility. To achieve thesis objectives, three main problems were considered that are presented in Chapters 3-6 respectively.

Chapter 2 focuses on reviewing the recent work related to trajectory planning for MAVs. In more detail, the finding of Chapter 1 include:

- The trajectory planning problem was broken down into a set of subproblems: free-space segmentation, motion model selection, initial waypoints identification, initial trajectory generation, continuous trajectory refinement, and receding horizon trajectory planning.

- For each subproblem, a descriptive in-depth review was carried out of how previous research has addressed those by presenting and evaluating various approaches to the considered subproblem.

The obtained finding was useful to understand what are the existing open research problems in the field of trajectory planning.

Chapter 3 deals with formulating a hard constraints-based planner for MAVs. The proposed hard constraints-based planner can track a given reference trajectory ensuring search space constraints in which changes in roll and pitch were not considered and the only change in yaw is considered. With such consideration, the simplified motion model-based was proposed, which helped to reduce the computational demands when calculating control policy. Subsequently, a proper investigation was carried out to predict residual dynamics, i.e., dynamics error between the hard constraints-based planner and the actual quadrotor, that arise when utilizing a simplified motion model when generating control policies. The residual dynamics latent distribution was modelled from training data as a Sparse Gaussian Process model. Afterwards, residual dynamics were added to the nominal motion model when formulating the nonlinear model predictive control. The

predictions of the residual dynamics model are invariant to both the time and geometric representation of the trajectory. Such invariance is present mainly due to the trained model having been trained only to fit residual dynamics. Finally, the augmented residual dynamics-based planner reduced the root mean square error of nominal model error by a factor of 1.5 and 2.5 without and with considering search space constraints, respectively, for trajectory tracking. More precisely, the results and contributions of Chapter 3 include:

1. Proposed a mathematical modelling approach based on Stochastic Linear-Quadratic Gaussian-(SLQG): a hard constrained-optimization problem with a simplified motion model to overcome computational demands with a short prediction horizon to obtain real-time performance. Furthermore, the proposed approach was designed based on two different parametrization techniques: Multiple-shooting (MS) and Direct Collocation (DC)

2. Proposed a mathematical modelling mechanism for residual dynamics learning: a Sparse Gaussian Process(SGP)-based model to alleviate the model residual between the simplified motion model and actual robot model. Moreover, realtime performance was enforced by running the complete algorithm on an onboard computer;

3. Constructed search space constraints directly from EDTM for identifying closest search space constraints along the NMPC prediction horizon.

Chapter 4 deals with developing a soft constraints-based planner that pushes reference trajectory towards the free space iteratively. The proposed approach was tested on various simulated and real-world environments, achieving long-range trajectory tracking. The soft constraints-based planner has a mean computation time of approximately 0.05s (20Hz), provided that tracking accuracy is less than 1m when there are no search space constraints. In more detail the results and contributions of Chapter 4 include

1. Developed a framework for reference trajectory tracking, ensuring search space constraints in which the soft constraints-based planner refines reference trajectory that ensure dynamic feasibility allowing the hard constraints-based

planner to generate a near-optimal control policy quickly at every planning iteration.

2. Proposed a fast approach formulated as a convex problem for pushing the reference trajectory away from occupied space, where a parallel version of Convex Decomposition [34] was implemented and a simplified approach for time allocation was proposed

Chapter 5 demonstrates the effectiveness in terms of accuracy and performance after introducing a soft constraints-based planner alongside the hard constraints-based planner. The proposed cascade planner comprises two planners: a soft constraints-based planner and a hard constraints-based planner. The soft constraints-based planner refines the initial reference trajectory when the trajectory goes through occupied space and lets the hard constraints-based planner calculates a near-optimal control policy. The cascade planner

— Implemented the cascade planner within the ROS framework focusing on performance improvements, e.g., code-level optimization techniques were used such as memory management, and the compiler is enabled with optimization level to -O2 to further reduce the execution time. The proposed cascade planner was designed to run in parallel as three separate threads: hard constraints-based planner, soft constraints-based planner, and main loop synchronize data among the threads.

— The initial reference trajectory using cubic B-splines must be provided as the external input to the proposed cascade planner. Apart from that, the depth sensor, e.g., r200, provides point clouds that are input to the Octomap server for building the environment map that is used for both hard constraints-based and soft constraints-based planners. The output of the cascade planner whose hard constraints-based planner calculates the desired control command to navigate the MAVs. PX4-enabled quadrotor was used for the simulation purpose. DJI M100 quadrotor was used for real-world experiments.

— The soft constraints-based planner addresses the reduction of the number of search space constraints that the hard constraints-based planner has to incorporate, the dynamic feasibility and smoothness of the reference trajectory,

and pushes the reference trajectory away from the occupied space. On the other hand, the hard constraints-based planner considers search space constraints and generates a near-optimal control policy.

- The Sparse Gaussian Process Regression helped to reduce the residual dynamics between desired and actual control commands.

Chapter 6 devotes to the efficiency testing of the developed cascade planner. Several types of experiments were carried out in simulated and real-world environments. All the simulated experiments, which are based on Gazebo, were conducted on a computer with (Intel(R) Core(TM) i9 CPU @ 2.50 GHz CPU and 16 GB RAM). The ROS framework was used to implement the complete framework.

- The first and second experiments were focused on reference trajectory tracking accuracy: without considering search space constraints in simulated vs real-world conditions and varying maximum allowed speed of MAV. In the simulated conditions, the average velocity error was 0.02± 0.01m/s, and in the real-world condition, the average velocity error was 0.07 ± 0.02m/s, for tracking a provided reference trajectory. The tracking accuracy decreases when increasing the MAV's speed because of the approximated nominal motion model.

- The third and fourth experiments were devoted to estimating the trajectory tracking accuracy of the hard constraints-based planner. The proposed hard constraints-based planner was compared with two other approaches: presented by [1] and by [2], which were targeted on the slow-speed maneuvers similar to the proposed approach. The same validation technique that [1] used for checking the behaviour in different environments was used. The experiment result showed that the proposed approach has a high success rate in terms of reaching the goal. However, the average run-time (0.1±0.05 s) was considerably higher compared to the others.

- The last two experiments were devoted to a comparative analysis of the proposed approach with a nominal model and with an augmented residual model. In the first experiment, three other methods: RRT* [4], a hard constraints-based planner [5], and FASTER [6], were used to validate the

proposed cascade planner (only considering nominal dynamics). The success rate was higher compared to the others. Moreover, the average run-time (0.03±0.01 s) has improved after introducing the soft constraints-based planner alone with the hard constraints-based planner. In the second experiment, the same experimental setup was used as in the first experiment. However, four different trajectory planners: sampling-based grid search followed by trajectory generation, NMPC-based replanner [4], hybrid planner (local and global) [7], and KinoJGM [8] were considered. All the selected planners have their advantages and limitations. Even after incorporating residual dynamics learning, the average run-time remains around 0.4 ± 0.01s and has the highest success rate compared to the other four approaches.

The essential properties of the proposed cascade planner are online trajectory refinement and near-optimal control policy generation in parallel in a horizon-based fashion, while only reasoning the surrounding environment without any prior knowledge of the search space. The proposed cascade planner was tested on various simulated and real-world environments, achieving long-range trajectory tracking. The hard constraints-based and soft constraints-based planners have mean computation times of approximately 0.06s (15Hz) and 0.05s (20Hz), respectively, on an NVIDIA Jetson Xavier NX computer, provided that tracking accuracy is less than 1m without considering search space constraints. The soft constraints-based planner focuses on real-time performance, whereas the proposed hard constraints-based planner ensures dynamic feasibility, and reference trajectory tracking accuracy, provided that the hard constraints-based planner and soft constraints-based planner have mean computation times 0.06s (15Hz) and 0.05s (20Hz), respectively. The source code and complete experiments are available at Github4

4 The source code and complete experiments - https://github.com/GPrathap/ trajectory-tracker.git

Limitation of obtained results

In spite of numerous essential advantages of the proposed cascade mathematical model, still has several limitations that are related improve accuracy and reduce average execution time. The most significant of them are presented below

- Formulates the hard constraints-based nonlinear optimization problem as a convex optimization problem.

- The proposed soft constraints-based optimization problem formulation is not guaranteed optimality, the topological trajectory generation approach can be applied to enforce the optimality.

- Introduces terminal constraint set to ensure the local optimality of the hard constraints-based problem formulation and improve recursive feasibility since the model predictive control paradigm does not guarantee optimality explicitly.

Such a change may further reduce execution time. However, it reduces the robustness of the cascade mathematical model because of the frequent infeasibility which does not occur when the formulation is in a nonlinear form.

Formulates the hard constraints-based nonlinear optimization problem as a convex optimization problem. The proposed hard constraints-based planner implementation was designed as a hard-constrained nonlinear optimization problem. However, sufficient control commands were generated using a simplified motion model in the proposed approach. However, utilizing approximated motion model in the planning stage produces a dynamical error (residual dynamics) between the planner and the low-level controller that is hard to estimate analytically. MPC is one of the robust methods for determining an optimal control policy imposing constraints seamlessly. In each iteration, MPC solves an optimal control problem for a given prediction horizon. Consequently, the first portion of the optimal policy is applied to the system. Since the optimal policy calculation procedure has to compute in each step, MPC is computationally expensive. However, for high-speed operations, it is not sufficient to control commands using a simplified motion model. Therefore, it is necessary to consider external aerodynamic effects, e.g., wind, that is applied on the quadrotor in addition to other constraints: system dynamics and search space constraints. However, several studies related to agile operations [136-140] do

not consider such effects, which are very difficult to incorporate when modelling system dynamics, except approximating the quadrotor dynamics with simplified motion model [14]. Even if those effects are incorporated, the necessary external aerodynamic effects are difficult to obtain due to high-computational demands that leverage real-time performance. In other words, model complexity is constrained by the computational capabilities of the onboard controller.

Employing a sophisticated system dynamical model in the planning stage is computationally expensive, which makes it hard to obtain real-time performance since search space constraints also have to incorporate when formulating the reference trajectory tracking problem. To overcome such computational demands, an approximated motion model is highly desirable in the planning stage. However, such a model has low maneuverability due to the low expressiveness of the dynamics of the system. To improve the expressiveness of the dynamical error (or residual dynamics), which occurs repeatedly, residual dynamics are incorporated such that the approximated model works its desired behaviour. However, these approximations and assumptions are difficult to anticipate for high-speed maneuvering. The proposed cascade planner can track a given reference trajectory considering search space constraints in which changes in roll and pitch were not considered and the only change in yaw is considered. With such consideration, the simplified motion model-based was proposed, which helped to reduce the computational demands when calculating control policy. Moreover, the problem was formulated as a constraints optimization problem. Thus, a sophisticated motion model is needed to capture the necessary dynamical effects to obtain appropriate high-speed operations. On the other hand, when using a sophisticated motion model the computational demand increases. To reduce the computational demand, the objective function should be linearized and formulated as a convex optimization problem in which both objective and constraints must be designed as convex. These design considerations are further reduced to guarantee a fast reaction time to handle dynamical search space constraints, which will improve reliable navigation in constrained environments.

The proposed soft constraints-based optimization problem formulation is not guaranteed optimality, the topological trajectory generation

approach can be applied to enforce the optimality. The key objective of the proposed soft constraints-based planner was to develop a continuous optimization-based refinement of the reference trajectory to 'push it out' of the occupied space in the global phase for multi-rotor aerial vehicles in unknown environments. Hence, optimality is not guaranteed, but rather try to improve the feasibility, which helps the hard constraints-based planner to calculate control policy quickly, i.e., the hard constraints-based planner's execution time is quite low when there is a fewer number of search space constraints. The proposed soft constraints-based planner was formulated as a convex optimization problem for finding the pushed control points, which is not optimal but it ensures that pushed control points are within the free space. However, instead of generating a signal trajectory if the planner is capable of generating multiple trajectories that are enforced kinodynamics of the vehicle and obtain the optimal trajectory out of those generated trajectories. However, this process has to carry out nearly 20 times per second to have a real-time response to environmental changes. If there are dynamical search space constraints, the trajectory which was calculated in the previous iteration no longer be optimal, i.e., free space can be occupied by dynamical search space constraints. Thus, optimality analysis of the soft constraints-based planner is hard to perform. Therefore, the proposed soft constraints-based planner ensures trajectory feasibility. However, these changes lead to smooth maneuvering.

Introduces terminal constraint set to ensure the local optimality of the hard constraints-based problem formulation and improve recursive feasibility since the model predictive control paradigm does not guarantee optimality explicitly. The terminal cost plays a significant role in terms of the stability of the system locally and globally. The local stability can be estimated by, e.g., Lyapunov's analysis compared to global stability. In addition to terminal cost, terminal constraints for states should be enforced, which is quite computationally challenging for real-time applications. Moreover, enforcing terminal constraints is even more difficult for non-linear dynamics. Consequently, stability and feasibility tend to improve for the longer receding horizon, which is quite challenging due to computational demands. Thus, in most of the practical

applications, terminal constraints are not enforced into the optimization procedure as similar to the proposed hard constraints-based planner. Furthermore, classical MPC lacks recursive feasibility. Several varieties of MPC have been proposed to address processing issues to a certain extent. In [34], MPC-based trajectory planning approach was proposed, ensuring both the local and global optimality. However, none of the aforesaid approaches formally guarantees stability and safety. Lyapunov's analysis can be applied to confirm the local stability. Moreover, the terminal constraints set [95] can be incorporated. However, those measures are time-consuming, which directly affects the real-time performance [96]. A set of CBFs was proposed for improving real-time performance without affecting the system stability in [97-99]. Recently, reference governors-based techniques were proposed in [100; 101], enforcing safety constraints. It is natural that designing a path planer is followed by the actual controller to maneuver MAV. In such approaches, a reference governor can be used to handle the stability and constraint satisfaction separately to ensure system stability [102].

The above approaches are employed to estimate optimal control policy for safe navigation while imposing stability either using Lyapunov functions or reference governors. On the other hand, Li et al. [103] proposed to obtain an optimal control policy using a SDDM. They have modelled the system dynamics as a linear, timeinvariant as x = Ax + Bu, where u indicates the control input. System state, i.e., x := (p(£),y(£)), consists of two parts: p and y, where p(t) denotes the quadrotor position at a given time t and y(t) describes the higher-order terms, e.g., velocity, acceleration, etc. In the latter work, quadratic norm was utilized to represent the error between robot position and search space constraints. The quadratic norm is defined as ||p||fl := \JpTRp, where R is a symmetric positive matrix. ] is fully determined by the MAV heading direction at a given time instance as follows:

f oil + (ci - c2) Mi, i f^z = 0 R[^z] = 1 ( 1 ^ №.II2, J *z= , (5.1)

I c11, otherwise

where both c1 and c2 are predefined scales such that > c1 > 0; this process is called the SDDM, trajectory will be bounded incorporating SDDM information.

Since quadrotor dynamics is linear, a reference governor [101] is introduced to maintain safety and stability. Thus, introducing a terminal constraint set to ensure the hard constraints-based planner optimality and improve recursive feasibility is quite a challenging task, when it comes to real-time applications.

Nevertheless, these limitations will be the focus of future investigation.

Future investigation and perspectives

To generalize the obtained results, it is required to address the above-listed problem: formulate the hard constraints-based planner as a quadratic problem rather than a nonlinear problem, the proposed soft constraints-based planner is not guaranteed optimality, and improve both the hard constraints-based planner optimality and recursive feasibility since the model predictive control paradigm does not guarantee optimality implicitly. Hence, it is reasonable to continue research in several directions. Future work will focus on the following aspects.

- Formulates hard constraints-based planner as a quadratic problem rather than a nonlinear problem.

- The topological trajectory generation approach for the soft constraints-based planner

- Introduces terminal constraint set to ensure the hard constraints-based planner optimality and improve recursive feasibility since the model predictive control paradigm does not guarantee optimality implicitly.

- Investigate model-free deep reinforcement learning-based techniques such as Proximal Policy Optimization and Asynchronous Advantage Actor-Critic Algorithm to overcome the limited expressiveness of the simplified motion model [195; 196]

On most occasions, paths which are obtained by planning techniques are suboptimal. Hence, the initial trajectory that is generated based on the initial path is to be refined, ensuring dynamic feasibility for controlling the MAV. Various approaches can be applied for trajectory refinement. However, enabling recursive feasibility, incorporating terminal constraints and convergence to the desired state are the

utmost importance considerations to be contemplated throughout the process. LQR and MPC are the two most popular approaches that are being used for receding horizon planning. LQR is applied for linear systems, whereas iLQR and DDP are applied for non-linear system. Both in LQR or iLQR, OCP is defined as an open-loop control problem. On the other hand, MPC is designed as a close-loop OCP. In other words, OCP is seeking actions knowing the behaviour of the surrounding environment.

In the context of optimal trajectory planning, simultaneously computing optimal control policy, which is required to respond to unknown, sudden disturbances, and handling kinematics as well as system dynamics (i.e., satisfying velocity and acceleration constraints) yields a challenging problem, especially for quadrotors. While geometry-based path planning techniques [90; 91] ensure the asymptotical optimality of a path, they however do not consider quadrotor dynamics. But, it is essential that the generation of an optimal control policy ensures dynamic feasibility. So, in [92; 93], LQR was incorporated into path planning, by which both dynamic feasibility and local optimality were guaranteed. However, local optimality does not necessarily yield global optimality [94]. In [29; 34], a set of motion primitives was used to find feasible trajectories ensuring both global and local optimality. When dealing with unknown disturbances, MPC is a more robust technique than LQR. In [34], MPC-based trajectory planning approach was proposed, ensuring both the local and global optimality. However, none of the aforesaid approaches formally guarantees stability and safety. Lyapunov's analysis can be applied to confirm the local stability. Moreover, the terminal constraints set [95] can be incorporated. However, those measures are time-consuming, which directly affects the real-time performance [96]. A set of CBFs was proposed for improving real-time performance without affecting the system stability in [97-99]. Recently, reference governors-based techniques were proposed in [100; 101], enforcing safety constraints. It is natural that designing a path planer is followed by the actual controller to maneuver MAV. In such approaches, a reference governor can be used to handle the stability and constraint satisfaction separately to ensure system stability [102].

Most of the recent optimal trajectory planning techniques [1; 2; 22; 32] were formulated as GTO in which optimization problem was designed as a non-linear form. The gradient descent is performed with respect to each parametrization index of Г to minimize the difference, i.e., Г^+1 — r¿. Hence, Г^+1 can be determined by solving the following optimization problem as given in [67; 116].

Г«+1 = argmin J (Г0 + (J (Г) — J (Гг))т V J (Г») + n 1|Г — Г\\2М , (5.2)

Г 2

where M is a weighting matrix and n is a regularization parameter. GTO is rather popular due to its ability to deform ineffability trajectory segments, low memory requirement and high throughput. Despite having the listed advantages, GTO can not avoid the problem of a local minimum. STOMP [114] is one of the early techniques proposed to address the local minimum problem. STOMP is based on the gradient-free technique. However, STOMP is unable to obtain real-time performance. Besides STOMP, the local minimum problem has been addressed by various recent works. Yet, this remains an open problem to be solved. Zhou [117] proposed a method, i.e., PGO, for overcoming the local minimum problem by generating topologically distinct paths and doing parallel optimization. Furthermore, various solvers can be utilized for solving optimization problems, including BOBYQA [118], L-BFGS [7; 119], ACADO [120], SLSQP [121], Proximal Operator Graph Solver (POGS) [122; 123], SQP and MMA [124]. Shravan et al. [54] proposed a trajectory optimization technique in a distributed setup in which the researchers evaluated their formulation with several solvers. According to their observations, BOBYQA is faster compared to BFGS and SLSQP, while MMA yielded a similar performance to that of BOBYQA. In [125], L-BFGS was proposed for finding the shortest path in real-time; in this research effort however L-BFGS does not guarantee optimality, only feasibility is enforced. MPCC [126] yet another proposed method for fast trajectory optimization in real-time. Moreover, Mathieu and Nicolas [127] proposed a SQP-based trajectory generation approach for carrying augmented loads. The intuition behind selecting SQP over other solvers is due to its super-linear convergence and ability to handle non-linear constraints within milliseconds.

In recent years, MAVs-related manifestations, e.g., trajectory tracking, exploration, expanding the range for venturing out computationally expensive techniques in various disciplines, including agriculture, aerial photography, and crop monitoring [197], [198]. Low-speed operations, in general, are preferred over high-speed operations in executing such demanding tasks due to task complexity. An example of such low-speed maneuver need is trajectory planning in a constrained search space. Therefore, the kinematic modelling of a quadrotor was considered since the scope of this work is for low-speed maneuvers without consideration of dynamic effects (external or/and internal). Similar assumptions were used in [199] for weed detection in crops, which showed the necessity of low-speed maneuvering. On the other hand, several studies have been carried out for high-speed maneuvers [137], [138], [139] with consideration of residual dynamics estimation, while most of them struggle with high-computational demands for on-board processing and difficulty in maneuvering in cluttered environments. The proposed framework can be adapted for high-speed maneuvers as well. However, it requires adding a behavioural planner that switches between planners according to the requirements, for example, in the cluttering environment we can switch to the low maneuvers while keeping the same control command generation. That will give us several benefits compared to existing solutions, e.g., residual dynamics models can be trained separately for low-speed maneuvers and high-speed maneuvers. Hence, the proposed solution can be adapted to general control with high and low-speed maneuvers introducing two or more operation modes and switching between them based on the external conditions.

Since Space Gaussian Processing is a non-parametric model, it does not require any parameter tuning. However, it is necessary to provide a proper training set that can capture the whole distribution of the latent space, i.e., residual dynamics. The simulated experiments of the proposed approach were carried out with an IRIS quadrotor with a PX4 flight controller. The real-world experiments were carried out with a DJI M100 with an A3 flight controller. In both cases, the initial step was to fine-tune the hyper-parameters of the PD regulator. This process is required before running the proposed cascade planner. Suppose the proposed approach is deployed

in another MAV. In that case, it is required to fine-turn the listed parameters since the high-level planner does not know any information about the low-level controller. However, this is a one-time process. Recently, auto PID tuning approaches were proposed [200]. Hence, it is required to investigate these aspects to improve the parameter tuning of the proposed approach.

Future extensions may include improvement of the hard constraints-based planner considering the aforementioned accepts. Also, the proposed approach is not limited to MAVs but also to other autonomous vehicles. Another possibility is to investigate model-free deep reinforcement learning-based techniques such as Proximal Policy Optimization and Asynchronous Advantage Actor-Critic Algorithm to overcome the limited expressiveness of the simplified motion model [195], [196]. Future work will focus on these directions.

Список литературы диссертационного исследования кандидат наук Кулатхунга Мудийанселаге Гисара Пратхап Кулатхунга, 2024 год

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