# Real-Time First Order Guidance Strategies for Trajectory Optimization in UAVs by Utilizing Wind Energy

## Abstract

This paper presents real-time guidance strategies for unmanned aerial vehicles (UAVs) that can be used to enhance their flight endurance by utilizing insitu measurements of wind speeds and wind gradients. In these strategies, periodic adjustments would be made in the airspeed and/or heading angle command for the UAV to minimize a projected power requirement at some future time. In this paper, UAV flights are described by a three-dimensional dynamic point-mass. Onboard closed-loop trajectory tracking logics that follow airspeed vector commands are modeled using the method of feedback linearization. A generic wind field model is assumed that consists of a constant term plus terms that vary sinusoidally with respect to the location. To evaluate the benefits of these strategies in enhancing UAV flight endurance, a reference strategy is introduced in which the UAV would seek to follow the desired airspeed in a steady level flight under zero wind. A performance measure is defined as the average power consumption both over a specified time interval and over different initial heading angles of the UAV. A relative benefit criterion is then defined as the percentage improvement of the performance measure of a proposed strategy over that of the reference strategy. Extensive numerical simulations are conducted. Results demonstrate the benefits and trends of power savings of the proposed real-time guidance strategies.

## 1 Introduction

Because of the generally light weights and/or small sizes of unmanned aerial vehicles (UAVs), wind can play an important, sometimes crucial, role in their flight endurance and performance. Given the ubiquitous nature of the wind, it is highly desirable to devise UAV flight strategies that enable them to benefit from the wind.

However, a main challenge in utilizing wind energies for practical UAV flights is the need to obtain accurate and timely wind information in real time. Because UAVs may travel to remote regions where ground support systems are not readily available and typically regional wind field information is unknown, airborne measurements of the local wind information are essential. Therefore, an interesting problem of UAV wind utilization is to develop onboard guidance and control strategies that can take advantage of wind energies based on airborne measurements (or estimates) of the local winds.

This paper presents such real-time guidance strategies in which optimal adjustments are made to the airspeed and heading angle commands to minimize a projected power consumption at some future time to prolong a UAV flight, based on the current local wind conditions. The onboard feedback control system then tracks these modified commands. This process is repeated periodically.

There are pioneering works in the area of UAV flights utilizing wind energies. The developments and flight tests of practical guidance strategies for detecting and utilizing thermals by Allen[1, 2] and Edwards[3] have illustrated the feasibility of these concepts. Boslough[4] demonstrated the benefits of utilizing wind gradients through dynamic soaring using radio-controlled UAVs. Patel and Kroo[5] studied the effect of wind in determining optimal flight control conditions under the influence of atmospheric turbulence. Langelaan and Bramesfeld[6] studied how to exploit energy from high frequency gusts in the vertical plane for UAVs. Wharington[7, 8] presented methods for learning the wind patterns, based on local sensing and an appropriately selected reward function, and to fly most efficiently. Pisano[9] investigated the gust sensitivity on the UAV dynamics as a function of aircraft size. In addition, Chakrabarty and Langelaan[10] presented a method for minimum energy path planning in complex wind fields using a predetermined energy map. Lawrence and Sukkarieh[11] developed a framework for an energy-based path planning that utilizes local wind estimations for dynamic soaring. Rysdyk[12] studied the problem of course and heading changes in significant wind conditions. McNeely[13] and et al. studied the tour planning problem for UAVs under wind conditions. McGee and Hedrick[14] presented a study of optimal path planning using a kinematic aircraft model.

Dynamic optimization methods have also been used to determine the full potential benefits of wind energy utilization when a regional wind model is known. Sachs, Knoll & Lesch[15] et al. studied optimal glider dynamic soaring in a wind gradient. Zhao and Qi[18, 17, 16] showed that under appropriate conditions with a full knowledge of the wind field in a region, a UAV can greatly enhance its endurance by properly utilizing wind energies. In addition to favorable wind patterns, such as wind gradient and thermals, Zhao[19] recently showed that even downdraft wind can be utilized to improve UAV performances. These results indicate that the utilization of wind energies for enhancing UAV flights is highly promising. Furthermore, dynamic optimization methods have also been applied to glider flights in winds[20, 21, 22]. In addition, Mueller, Zhao & Garrard[23] studied optimal airship ascent flights by utilizing wind energy. In these studies, the nonlinear dynamic optimization formulation typically requires knowledge of regional winds and an iterative solution process, and thus may not be feasible for generating real-time guidance strategies. Still, they are useful in understanding fundamental patterns of optimal UAV flights in winds and providing benchmark results that can be used to evaluate real-time wind utilization strategies. Approximate solutions may also be obtained for nonlinear dynamic optimization problems to derive real-time guidance strategies.

Compared with the dynamic optimization studies, the current paper presents real-time guidance strategies that use insitu wind measurements alone with no regional wind information, to reduce power consumptions. These strategies periodically adjust airspeed vector commands to take advantage of changes in the mean wind profile. Wind energies in the changing mean wind profile are generally of lower frequency compared with gust energies. As a result, the proposed real-time guidance strategies complement the previous works on real-time guidance and control methods that utilize gust energies.

In the current paper, only airspeed and heading adjustments are examined with a zero flight path angle command in order to evaluate the benefits of unconstrained guidance strategies. These strategies can be easily expanded to adjust the flight path angle as well, but this requires the incorporation of altitude boundary control in order to prevent the UAV from hitting the ground. Guidance strategies that need to respect flight constraints in altitude as well as over the horizontal region shall be reported later.

In the rest of the paper, three-dimensional point-mass equations are used to describe UAV motions in winds. Optimal adjustments for airspeed and heading angle are derived by minimizing a power consumption projected into the future. Models of closed-loop trajectory tracking are developed to follow airspeed vector commands, using the technique of feedback linearization. A generic wind pattern consisting of a constant term plus spatially varying terms is used to evaluate the average power consumption of the proposed strategies. In order to eliminate the impact of wind directions on the relative benefits of proposed strategies in UAV flights, the power consumption is averaged both over a specified time interval and over different initial heading angles of the UAV. This average is then compared with that of a reference strategy, in which the UAV seeks to maintain a steady level flight with the airspeed that would maximize the endurance in zero wind. Conclusions are drawn at the end.

## 2 Equations of Motion and Constraints

For the purposes of developing guidance strategies, UAV flights are represented by a dynamic point-mass model. The corresponding normalized equations of motion for a propeller-driven UAV are listed below[24], where the UAV mass is assumed to be constant.

### 2.1 Normalized Equations of Motion

In order to increase numerical efficiency in the simulation studies, the above equations of motion are normalized by specifying a characteristic speed and mass . We have

(1) |

(2) |

and

(3) |

where the normalized air-density, , represents the combined effect of air density () and wing loading () on UAV flights. Specifically, a smaller corresponds to a larger wing loading (a heavier UAV) and/or thinner air, where a larger represents a lighter UAV (with a smaller wing loading) and/or thicker air.

Using these normalizations, the normalized drag and lift become

(4) |

The normalized wind components are defined as

(5) |

where the normalized rates of wind speeds follow similar expressions as in Eqs. (7)-(9).

Then the set of normalized equations of motion are obtained as follows, where functional dependences of the wind terms are shown for convenience

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

Constraints on states and controls can also be expressed using normalized values.

## 3 Problem Statement

The main aim of this paper is to develop real-time guidance strategies to enhance the endurance of UAV flights based on insitu wind information. Ideally, if the regional wind information is completely known in advance, optimal flight planning can be used to determine UAV flight trajectories that minimize the total power consumption over a specified time interval, subject to various constraints. However in this paper, it is assumed that only wind information at the current location of the UAV at the current time is available. This information includes values of wind speeds as well as wind gradients.

In general, different guidance strategies may be grouped into three basic categories: action strategy, velocity strategy, and trajectory strategy. This paper studies velocity guidance strategies that can utilize insitu wind information to enhance UAV endurance. A propeller-driven UAV is assumed, for which maximum endurance corresponds to minimum power consumption[25]. Therefore, we seek to determine incremental adjustments in airspeed and heading angle to minimize the power consumption projected sometime into the future. Mathematically,

(12) |

subject to all applicable constraints. Then, the UAV will be directed to track and commands.

Once adjustments in airspeed and heading angle are obtained, it takes some finite time for the UAV to achieve the desired changes via closed-loop tracking. As a result, a projected power consumption at is used instead of the current power in Eq. (12).

## 4 Solution Strategies

A key to solving the above problem is to develop an expression for the projected power required at the time , based on values of the current trajectory state variables as well as the current wind information. Therefore, should neither be too large or too small. It needs to be larger than the typical settling time of inner closed-loop controls in order to ensure that any adjustments in airspeed and heading commands will have been achieved. At the same time, too large a would reduce the accuracy of power consumption projections using the current state and wind information.

In this paper, it is assumed that the UAV intends to maintain a level flight: , , and .

(13) |

From Eq. (6), we have

(14) |

Therefore, the projected power level at is given by

(15) |

It is assumed that by the time , any commanded changes in airspeed and heading angle will have been mostly achieved via closed-loop tracking. Therefore, the vehicle is basically in a steady state: and . We have,

(16) |

We now need to develop an expression for the term.

(17) |

### 4.1 An Expression For Projected Wind Rate

Because

(18) |

where . By substituting previous expression in Eq.(18), we have the expression given in Eq. (17), where

(19) |

In level flights with negligible vertical winds, , ,

(20) |

Because only insitu wind information is available, it is assumed that the current wind gradients shall stay constant over the immediate neighborhood around the current position of the UAV in the near future. This assumption shall be called the “constant wind gradient assumption”. Therefore, we obtain the final expression given in Eq.(21).

(21) |

With the constant wind gradient assumption over , we also have

(22) |

These expressions depend on , which depend on , , and , and reciprocally on the wind components over the interval. Therefore, we need to develop expressions of in order to complete the derivation of the projected wind rate expression.

### 4.2 Expressions for Position Changes

We now seek to develop expressions of that show their dependencies explicitly on the increments of airspeed and heading angle. After experimenting with different methods, the following expressions are obtained. From Eqs. (9) and (10), we have for ,

(23) |

Applying the trapezoidal rule[26] to integrate the above equations, with the assumption that both airspeed and heading angle will have achieved their commanded values at the end of the interval and the wind speeds are given by Eq. (22), we obtain

(24) |

Define,

(25) |

we have

(26) |

or

(27) |

where

(28) |

For a sufficiently small update time-, this expression shall always be nonzero; ensuring the existence of solutions for the position change expressions.

### 4.3 Guidance Algorithms

Based on the above derivations, we can now express the projected power consumption at as a function of the current command adjustments in airspeed and heading angle. Then, the problem of reducing future power consumptions is to determine and from Eq.(29) with corresponding constraints in Eq.(30).

(29) |

subject to

(30) |

where and are the maximum allowed incremental changes, and the initial state conditions required include

(31) |

As a reference strategy, it is assumed that the UAV follows a constant airspeed straight level flight. The airspeed is optimal in zero wind. In this case, the projected power expression in Eq. (16) becomes

(32) |

(33) |

This airspeed corresponds to the maximum endurance under zero wind in a steady level flight.

## 5 First Order Adjustment Strategies

Different algorithms can be used to solve the static optimization problem in Eq. (29). In this paper, first-order gradient algorithms are used to obtain solutions. In deriving a first-order gradient method, we approximate the projected power expression as

(34) |

where corresponds to zero commanded adjustments or .

### 5.1 Airspeed Adjustment Strategy

A first-order incremental airspeed adjustment strategy can be determined from Eq. (34) as

(35) |

where is introduced to avoid numerical difficulties in implementation, and is the adjustment stepsize. Expanding Eq. (16) using Eq. (20) leads to

(36) |

where the partial derivatives of wind and position changes with respect to the speed increment can be obtained from Eqs. (27) and (25) as

(37) |

and

(38) |

### 5.2 Heading Strategy

Similarly, the incremental heading change can be obtained from Eq. (34) as

(39) |

where could be expressed in compact form as

(40) |

and

(41) |

## 6 Simulation Evaluation

In the current paper, the Aircraft Dynamics is modeled by the previously described point-mass equations. The UAV tracking logic, contained within the Trajectory Tracking concept, is based on the method of feedback linearization as described below. In general, the Wind Estimation process represents sensors and algorithms for deriving estimates of the current wind states. The current paper seeks to focus on the development of algorithms for utilizing wind energies. It is therefore assumed that accurate wind estimates can be made and are available. Future work shall consider effects of errors associated with wind measurements and estimations.

### 6.1 Models of Closed-Loop Tracking

It is assumed that once optimal incremental adjustments (, ) are derived, the UAV would track these commands in their flights. Actual onboard trajectory control logics can be very complicated and can also vary from vehicle to vehicle. In this paper, the method of feedback linearization is used to develop models of actual onboard trajectory tracking logics. The point-mass dynamic model has three control variables: (or ), and . Therefore, we need to develop three closed-loop trajectory control models.

Use of the feedback linearization method starts with the specification of desired closed-loop dynamics. Specifically, a desired closed-loop airspeed tracking using thrust can be specified as a first-order system.

(43) |

Using normalized variables, the closed-loop thrust law can be determined from Eq. (6) as

(44) |

where is the normalization time. Similarly for the heading control using bank angle, we have

(45) |

which leads to

(46) |

Finally, tracking a commanded flight path angle using lift coefficient can be achieved with

(47) |

which results in

(48) |

Combining Eqs. (46) and (48), we obtain

(49) |

and corresponding is given by Eq.(50).

(50) |

In the above, the feedback gains can be selected to reflect typical closed-loop UAV control characteristics. In this paper, it is assumed that , , and , all in sec.

### 6.2 Guidance Algorithm Parameters

Performances of the proposed guidance strategies strongly depend on the following four parameters

(51) |

Ranges of their appropriate values are now estimated.

Similarly for the maximum heading angle change, we have