Energy-Optimal Unmanned Aerial Vehicles Motion Planning and Control Based on Integrated System Physical Dynamics

Multirotor aerial vehicles with electric-vertical-take-off-and-landing (eVTOL) capability are a technology with significant potential for a wide range of applications, with main advantages including superior maneuverability [1] and zero emission potential as an electrified transportation mode [2] compared with the conventional aircraft. Nowadays unmanned aerial vehicles (UAV)/drones are already seeing increasing military applications including intelligence, surveillance, and reconnaissance (ISR) missions, and civilian use cases such as aerial photographing, wireless communication, and monitoring of environmental conditions [3–5]. Furthermore, they have also been envisioned as a critical part of the future logistic and transportation networks to supplement the traditional ground transportation mode [6,7]. For example, decimeter-scale drones are under development for package delivery over air [8,9], and larger eVTOL aircraft are being considered to enable “flying cars” for urban air mobility (UAM) [10,11].

Despite the potential of revolutionizing the transportation and logistics networks, the UAV and UAM technologies also face critical obstacles, including a significant one in energy performance. Currently, small scale multirotor drones are subject to major constraints on flight time (typically up to approximately 30 min) and limited payload [4]. The energy requirement is similarly a major constraint for large-scale UAM applications [7], which requires more than 1 h of flight time, over 70 miles range, and high power during takeoff and landing [11]. Trajectory/motion planning and control, which is a widely studied topic in UAV literature [12–14], has been viewed as an effective approach to minimize energy consumption and improve the energy performance of the system and has received extensive research attention in literature. Many existing works exploit some of the underlying system dynamics [12,15,16] based on which the multirotor motion can be optimized to enhance energy efficiency. In particular, some papers focused on planning and control considering propeller aerodynamics, which govern the thrust and torque generation for propulsion. For example, the power consumption of a UAV was modeled based on the propeller blade element momentum (BEM) theory in Ref. [17], which was then combined with optimization algorithms to generate energy-optimal two-dimensional flight trajectories to reach a destination in Ref. [18]. The same model was also used in Ref. [15] for optimizing flight plans to avoid obstacles and minimize energy consumption in a simulated two-dimensional wind field. Another model, which takes into account the inflow momentum theory and rigid body dynamics of the UAV airframe, was described in Ref. [13] and used to evaluate and compare the energy performance of a UAV in level forward flight operations under various existing control approaches, including minimum acceleration, minimum jerk, and minimum snap control. Some other works investigated the dynamics of the motor and motor controller and exploited them for energy-efficient planning and control. In Ref. [12], a model was formulated to calculate the vehicle power consumption based on the motor and airframe rigid body dynamics and was then applied to generate trajectories for quadrotor hover-to-hover flight operation, minimizing either the energy consumption under a fix time budget, or flight time under a fix energy budget. This model has also been used to optimize energy consumption with either free or fixed end time in Ref. [19]. Another quadrotor model, which similarly considers motor and airframe dynamics, was introduced and used to generate energy-optimal trajectories in Ref. [20]. This approach was further expanded to incorporate wind effect and predict the change in battery voltage in Refs. [21] and [22]. In a similar work [3], rigid-body and motor dynamics were considered for trajectory optimization of a drone used for wireless communication, where communication performance was also included in optimization. There have also been efforts on accommodating the dynamics of the power source. For example, a UAV altitude controller has been designed in Ref. [23], which takes into account the battery state of charge (SOC). Additionally, energy-optimal paths for waypoint-to-waypoint operations were studied in Refs. [24,25] considering vehicle rigid body and motor dynamics, and battery state-of-health.

While these works have made important contributions toward improving the multirotor energy performance, they each only consider part of the governing multi-physics of the full vehicle. The complete dynamics include the subsystem dynamics of all related components, i.e., the aerodynamics of the rotor-propeller assembly, electro-mechanical dynamics of the motor and the electronic speed controller (ESC), electrical dynamics of the battery, and rigid body dynamics of the airframe, and more importantly, their mutual impact and coupling effect on energy performance. Negligence or oversimplification of the complete dynamics would cause incorrect projection of system behavior and lead to motion planning and control that is nonoptimal in performance or even unachievable/unsafe [26,27]. For example, in Ref. [27], rotor aerodynamics were demonstrated to have a significant impact on vehicle energy performance. Specifically, the energy-optimal velocity for steady forward flight, which is the target to track in various UAV control literature [28,29], was shown to be significantly overestimated when the rotor inflow dynamics were not considered. As a result, the energy cost per meter traveled is 67.8% higher than that under the true optimal velocity considering the rotor aerodynamics, which is determined by four tradeoff factors involving multiple subsystem dynamics. The battery subsystem dynamics are also noteworthy and can significantly affect vehicle-level propulsion performance [30,31]. Specifically, the rotor thrust and torque production will decrease when the battery voltage drops due to energy depletion, and the amount can be as much as 26% over the battery operation range [27,32]. This reduction in the available thrust can reduce the vehicle operational limit, e.g., maximum velocity and acceleration, potentially making it impossible to safely exploit energy-efficient maneuvers if the impact of battery dynamics is not considered in planning and control [26]. These results demonstrate the critical need for multirotor planning and control research based on integrated full system dynamics. It is also noted that individual subsystem dynamics do not affect the vehicle performance independently but rather through the coupling with each other. For example, the rotor inflow aerodynamics are coupled with the airframe rigid body dynamics, while the impact of battery voltage on propulsion could only be captured through the dynamics of the motor and motor controller [27].

To address the gap in the state of the art, this paper explores an energy-optimal trajectory planning and feedback control framework based on a system-level multirotor dynamic model. To the best of our knowledge, this work is the first to consider fully integrated subsystem dynamics for multirotor motion planning and control, with results validated by simulation and experimental testing and explained by correlating to the underlying multi-physics. As described in Sec. 2, the model used, which was previously developed in Refs. [32] and [27], includes propeller aerodynamics, motor and ESC electro-mechanical dynamics, battery electrical dynamics, and vehicle frame rigid-body dynamics. Incorporating each of these subsystems, as well as their integration, allows for comprehensive modeling of the energy dynamics of the complete vehicle. To demonstrate this capability, model-based analyses of key system characteristics affecting energy performance are provided in Sec. 2.6, which include (1) the relationship between rotor angular velocity and efficiency (in terms of thrust generated per Watt of input power), (2) the relationship between vehicle cruising velocity and energy cost per meter, and the governing tradeoff factors, and (3) the effect of wind on vehicle energy performance in constant-velocity forward flight. In Sec. 3, the full octorotor vehicle model is used for energy-optimal trajectory generation over a range of waypoint-to-waypoint flight operations, with a PID feedback control architecture developed for real-time trajectory following. Two baseline controllers commonly used for UAV control are also described for comparison. The developed planning and control framework is first tested in simulation, showing significant improvement in energy consumption over the baseline. Using the model-based energy dynamics analysis described previously, numerous behaviors are identified, which explain the substantial reduction in energy consumption achieved by the developed framework. In general, the framework enables holistic planning over the full duration of an operation, and efficiently balances the trajectories of multiple vehicle states along different motion directions (e.g., forward and vertical). Finally, experimental validation of the developed framework is performed using an octorotor platform under real-world operating conditions. The results show that the framework could achieve significant energy performance improvements over the baseline despite the presence of significant wind disturbances.

The remainder of this paper is organized as follows. First, the system-level multirotor model is described in Sec. 2, which also includes a discussion of key characteristics relevant to the energy performance of the vehicle. The model is then used for optimization of vehicle trajectories in Sec. 3, where a trajectory-following feedback controller is also described. The performance of the framework is compared to baseline controllers over a range of operations in simulation. Next, experimental validation is discussed in Sec. 4, with results presented and analyzed to demonstrate the performance of the developed planning and control framework under real-world testing conditions. Finally, main results and significant findings are summarized in the conclusion (Sec. 5).

In order to properly capture the complete physical dynamics of the multirotor UAV, a system-level model is used. This model considers the aerodynamics of the propeller-rotor assembly [33], the electro-mechanical dynamics of the motor and the ESC [34,35], electrical dynamics of the battery [36,37], and the rigid body dynamics of the vehicle [38], as well as the integration of each subsystem dynamics into the full model. The complete model derivations, as well as the parameterization process and results, can be found in Ref. [32]. An overview of the model and key characteristics that affect the energy performance of the vehicle is provided in this section.

The integration is performed from the base of the blade $R0$ to 97% of the tip $R$ (instead of 100% to approximate the tip loss) and averaged over one full revolution. These equations have also been simplified using the small-angle approximation and the conditions $upr≪upl$ and $dD≪dL$⁠, which have been validated in Ref. [32].

The last equation can be solved iteratively to calculate $vi$ using the value of $T$ found from the blade element theory.

To calculate the angular velocity $ω$ and the current draw $Ib$ for each rotor, a model of the electromechanical dynamics of the motor-ESC system is needed. The ESC model calculates $Ib$ as well as the motor input voltage $Vin$⁠, which is used in the motor model along with motor/propeller torque $Q$ from Sec. 2.1 to determine $ω$ and the motor input current $Iin$⁠. The inputs to the ESC model are the battery voltage $Vb$⁠, the pulse-width modulation (PWM) command, and (the looped-back) $Iin$⁠.

where $ηESC$ is the nonlinear ESC efficiency as a function of $Vb$ and PWM. In this model, other motor losses, such as friction and magnetic losses, are included in the motor zero-load current $I0$⁠, which is the current drawn when the torque load is zero [40].

A battery model is needed to capture the evolution of battery voltage and internal states over time driven by the current (power) load. Battery voltage serves as an input to the ESC, affecting the ESC output voltage to the motor and eventually the propulsion performance including thrust, torque, and rotor speed. In this paper, battery is modeled using an equivalent circuit model (ECM) [30] as shown in Fig. 2(b), which is one of the most popular approaches for modeling macroscopic battery behavior [31].

The resistance $Rk$ and capacitance $Ck$ of the RC pairs often vary with SOC and temperature [30]. The optimal number of RC pairs, $n$⁠, are typically determined by balancing the model fidelity and complexity. From parameterization data, $n=3$ was found to accurately model the battery used in this work [27]. A thermal submodel can be conveniently integrated with the equivalent circuit model to capture the battery temperature variation during operation, and the effects on battery behavior if necessary [41,42].

Finally, the rigid body dynamics of the airframe are considered to model the UAV motion using the thrusts and torques from the propeller model. Two-dimensional dynamics are described here, which are adequate for the motion considered in this paper, while standard equations for three-dimensional dynamics can be found in existing literature [38].

where $xj$ is the x position of the rotor relative to the vehicle center of mass.

Based on preliminary experiment results, it was found that the effect of rotor inflow was slightly overestimated for the test vehicle, which will be described in Sec. 4. This is likely due to the inflow interference between rotors, as well as the obstruction of air flow by the vehicle airframe. It was found that a scaling factor can be applied to the inflow velocities to correct for these effects with sufficient accuracy. Specifically, $vx$ is corrected with an 80% scaling factor, while that for $vz$ is 70%. These values were calibrated based on test data, and found to compensate for majority of the error between the predicted and measured thrust.

With each subsystem dynamics defined, the overall system-level model can be integrated as shown in Fig. 4. The control inputs are the PWM commands for each motor, which instruct the ESC to regulate the motor input voltage as a fraction of the battery voltage. The motor responds with a current draw and a rotational speed, which also depends on the torque load from the propeller. The motor rotation drives the propeller to generate the torque and thrust per blade element theory. Thrust and torque are then used in the rigid body dynamics model to calculate the motion of the UAV, which affects the horizontal and vertical velocities of each rotor for both blade element and momentum theories. The torque is looped back to the motor to determine the motor speed and current, which is further looped back to ESC to determine the current drawn from the battery. Finally, the voltage output by the battery varies in response to the total current load, which influences the voltage output from each ESC to the corresponding motor, completing the loop of coupled dynamics.

Based on the model equations, some critical relationships can be identified that will have a significant impact on the energy-efficient trajectories to be discussed in Secs. 3 and 4. First, the rotor efficiency, defined as thrust generated per unit input power, decreases as the rotor angular velocity $ω$ increases as demonstrated in Fig. 5. It can be derived from Eqs. (2) and (3) that both $T$ and $Q$ are approximately proportional to the square of $ω$ [32]. Consequently, the mechanical power output from the motor to the rotor, which is computed as $Qω$ and hence cubic of $ω$⁠, increases with $ω$ more rapidly than thrust. This effect is especially pronounced at high angular velocities, where a small increase in thrust demand can cause a disproportionately larger increase in power consumption, hence reducing the energy efficiency.

Second, for steady-state cruising operation, the relationship between the forward velocity and energy used per meter traveled, which measures the energy efficiency, can be derived based the model as shown in Fig. 6. The nonmonotonic trend is governed by four tradeoff factors, which are the results of the interplay between different subsystem dynamics. First, energy efficiency initially increases rapidly with velocity, because of the reduction in time traveled per meter. Second, thrust demand increases dramatically with horizontal velocity, resulting in higher energy use. This is because the resistant body drag grows quadratically with velocity, and therefore more thrust needs to be generated to provide the required horizontal thrust to balance the body drag at a higher pitch angle. Third, at low pitch angles (under low velocity), forward velocity primarily increases $vx$ according to Eq. (14), which generally has a positive effect on rotor efficiency. Specifically, higher $vx$ will increase the root-mean-squared $upl$⁠, which raises the thrust generation and the total lift-to-drag ratio $L/D$ over a full rotor revolution. Finally, as pitch angle further increases at high velocities, $vz$ instead grows rapidly, causing reduced performance and increase in energy cost. This is because higher $vz$ (and thus, $upr$⁠) will decrease $α$ and $cl$⁠, and hence the lift-to-drag ratio, reducing the thrust generation and rotor efficiency. Balancing the four tradeoff factors, the minimum energy cost per meter is shown to be 76.6 $J/m$ at the optimal forward velocity of 12.5 $m/s$⁠. These factors also determine the maximum velocity the vehicle can sustain during steady forward flight, i.e., 23.7 $m/s$⁠, above which the rotors are unable to generate sufficient thrust to maintain altitude while overcoming body drag. Change in battery voltage can significantly affect this velocity limit, as the maximum available thrust would decrease as the battery energy depletes [32]. While Fig. 6 is generated at a typical operating voltage of $Vb=23$V, over the full operating range from 21 V to 25 V, the maximum cruise velocity would vary between 22.1 $m/s$ and 25.3 $m/s$⁠.

Finally, the effect of wind on energy performance can be analyzed, which is also shown in Fig. 6. Although in this work wind speed is not assumed as known for trajectory generation and UAV control, understanding of wind effect could help explain the experimental results, which are inevitably affected by wind. Considering only the horizontal wind, a hypothetical or measured wind speed can be easily incorporated into the model as an input. This is achieved by including the wind speed in $X˙$ to calculate the body drag in Eq. (13) as well as $vx,j$ and $vz,j$ in Eq. (14). With this correction, results are calculated using the airspeed instead of the ground speed. The dashed curve in Fig. 6 shows the adjusted efficiency–velocity relationship under 5 $m/s$ of wind opposite vehicle motion (headwind), and the dash-dotted curve shows that under wind along vehicle motion (tailwind). Based on the tradeoff factors previously described, the effect of wind is intuitive at high velocities, where wind opposing the direction of vehicle motion increases energy consumption by increasing the drag resistance, forward pitch angle, and $vz$⁠. Similarly, wind in the direction of vehicle motion has the opposite effect. Consequently, the energy-optimal velocity and energy cost per meter vary significantly under wind as shown in the figure. With a headwind of 5 $m/s$⁠, the minimum energy increases 59% to 121.6 $J/m$ at 9.3 $m/s$⁠, while a tailwind reduces this energy cost by 29.4% to 58.8 $J/m$ at 16.3 $m/s$⁠. It should also be noted that at low velocities, headwind could actually reduce energy consumption, as it increases $vx$ and hence thrust production while the growth in drag resistance is minimal. Again, the opposite effect is observed for the tailwind, as the wind in this case reduces $vx$⁠. However, low-velocity behaviors are less significant, particularly as the energy-optimal velocities are primarily affected by the high-velocity behaviors previously discussed.

The developed system-level dynamic model of the UAV is used for model-based optimization of vehicle trajectories in this section. A feedback control architecture is also developed for trajectory following, and a baseline controller is described, which will be used for comparison. In this section, the results are generated and evaluated in simulation, and used to discuss key features and behaviors of energy-optimal UAV operation.

for projecting the state evolution.

are first obtained by resolving the equations in Sec. 2 that are related to the propeller, motor, and ESC dynamics. By using the tables, optimized solutions can be found without repeatedly resolving the loop in Fig. 4 at each time-step of the iterative optimization process, significantly reducing the computational intensity of the problem while still providing accurate results.

In this work, solutions are examined for a variety of operations. These operations include forward flights with distance of 50 m, 70 m, and 100 m, as well as diagonal climbing flights with the endpoints set at 50 m, 70 m, and 100 m ahead of and 10 m and 20 m above the starting position. Due to the computational load of solving the optimization problem, generating optimized trajectories on demand for real-time vehicle operation may not be feasible, especially if using a vehicle flight controller with limited processing power. Consequently, optimized solutions are determined off-board for the selected operations prior to testing, and the target states are interpolated from these solutions each time the control signal is updated during operation, providing accurate trajectories with minimal real-time computational load. The development of a more flexible and dynamic approach using polynomial approximations of the optimized trajectories is the subject of ongoing work [43]. Detailed analysis of the features and energy saving performance of the trajectories will be provided in subsequent sections.

In order to implement the generated target trajectories for real-world UAV operation, a PID control architecture is used for trajectory following as shown in Fig. 7. This control architecture consists of a high-level PID controller and a low-level one.

The high-level controller emulates a common manual flight control mode by using a set of PID controllers to generate commands (denoted with the subscript $C$⁠) for the pitch angle $Θ$⁠, roll angle $Φ$⁠, yaw rate $Ψ˙$⁠, and vertical velocity $Z˙$⁠. The commands are generated based on the error between the optimized target trajectories for these vehicle rigid-body states, denoted with the subscript $t$⁠, and the real-time feedback of the corresponding states. For instance, forward motion is controlled using the pitch angle command $ΘC$ generated based on the target values for $X$⁠, $X˙$⁠, and $Θ$⁠. The trajectories of $Θ$ and $X˙$ are needed due to their significant impact on energy performance, and are supplemented with the $X$ trajectory to ensure that the vehicle reaches and stabilizes at the correct end state. The roll angle command $ΦC$ is similarly regulated using the target $Φ$⁠, $Y$⁠, and $Y˙$ to control horizontal movement perpendicular to the forward axis. As vertical and angular accelerations can be controlled more directly than horizontal motion, the vertical velocity command $ZC˙$ is regulated based on only $Z$ and $Z˙$⁠, while the yaw rate command is regulated based on $Ψ$ and $Ψ˙$⁠.

The low-level controller contains another set of PID controllers, which receive the commands from the high-level controller and regulate based on the feedback of the corresponding vehicle states. These controllers regulate the acceleration of the pitch, roll, and yaw as well as the throttle command $TC$⁠. Based on these commands, a mixer equation is used to calculate the PWM for each rotor ESC, which are the lowest-level actuation inputs applied to the UAV. The throttle command dictates the average PWM across all rotors, which is then modified for each motor to provide the desired angular accelerations of each rotor. Each PID controller in this architecture is calibrated to closely follow the optimized state trajectories with good disturbance rejection.

As benchmark for quantifying the improvement in energy performance, commonly used way-point based controllers are also designed and implemented for comparison. These baseline controllers are constructed using the Auto Mode of the ArduPilot Mission Planner software. In this flight mode, the target horizontal and vertical velocities are set to be linearly proportional to the remaining distance along each direction. The yaw angle is controlled to orient the vehicle toward the waypoint, while the pitch and roll angles are regulated to provide the horizontal acceleration required to match the desired velocity. To ensure that the vehicle remains stable and the commands are within the operational range of the vehicle, user-defined constraints are imposed on the velocity and acceleration. Specifically, the vertical velocity, vertical acceleration, and horizontal acceleration constraints were set at 5.0 $m/s$⁠, 1.0 $m/s2$⁠, and 4.5 $m/s2$⁠, respectively, and two horizontal velocity constraints, namely, 18.0 and 12.5 $m/s$⁠, were tested. The higher velocity limit (⁠$18m/s$⁠) of the first controller, which is referred to as the high-velocity baseline (HVB) controller, allows the vehicle to reach the endpoint as quickly as possible, while avoiding the risk of failing to maintain altitude at higher velocities under adverse wind conditions. The velocity limit (⁠$12.5 m/s$⁠) of the second controller, which is referred to as the low-velocity baseline (LVB) controller, is the energy-optimal cruising velocity identified previously from Fig. 6 under no wind condition. With this limit, the slower baseline controller is expected to complete operation with lower energy consumption than the faster baseline controller, which operates under more aggressive but less energy-efficient velocity.

In order to evaluate the performance of the proposed control framework under nominal conditions, the optimal trajectory following and baseline controllers are first implemented and tested in simulation based on the developed multirotor system model. The time and energy required for each controller to complete a series of operations, with target final horizontal position $Xf$ ranging from 50 to 100 $m$ and vertical position $Zf$ from 0 to 20 $m$⁠, are obtained and summarized in Table 1. It should be noted that it could take a long time for the vehicle to come to a complete stop at the exact end position, due to overshoot and oscillation. Therefore, the results are calculated at the moment the vehicle moves within a 3 $m$ radius of the target end position. Since the cutoff radius is relatively small and applied equally to each controller, the effect of truncating the operations is minimal and similar, allowing for a fair comparison.

As shown in Table 1, the optimized-trajectory-following (OTF) controller reduces the energy consumption significantly over the baseline controllers in all cases, while the low-velocity baseline consistently provides a slight improvement over the high-velocity baseline. Specifically, the OTF controller uses an average of around 10.7% and 8.8% less energy than the HVB and LVB controllers, respectively. The optimized trajectories demonstrate distinct patterns from those under the baseline controllers, as shown in Fig. 8, which compares the evolution of key vehicle states under each controller and in two sample operations. Specifically, the first column of the subplots, i.e., Figs. 8(a)–8(e), shows the trajectories of X (horizontal) position, X velocity, pitch angle, Z (vertical position), and cumulative energy consumption for the $100 m$ forward flight, and the second column, i.e., Figs. 8(f)–8(j), shows those for the diagonal flight with $50 m$ horizontal and $20 m$ vertical motion. In general, the optimized trajectories can be approximately divided based on the forward velocity profile into three segments as shown in Fig. 8(b), including an initial period of acceleration to the peak forward velocity, a middle period of cruise and gradual deceleration, and a final braking period when the vehicle rapidly decelerates to stop at the endpoint. The optimized trajectories are shown to include behaviors that improve energy performance in each of these flight segments, as well as effectively balancing the simultaneous horizontal and vertical motion.

First, the improvements in the initial acceleration period are achieved by reaching a steep (negative) pitch angle at the start of the operation to provide high forward thrust. Note that from Eq. (13), a negative pitch angle will result in positive forward acceleration due to the orientation of the body-frame axes as shown in Fig. 3. As a result, the forward velocity peaks early, reducing the total time required to reach the endpoint. This peak velocity is lower in operations with a small horizontal component, where high forward velocity is not required or cannot be reached. By comparison, the baseline controllers pitch forward much slower, as they cannot be well-calibrated considering the vehicle system dynamics. Consequently, in relatively short flights, the vehicle will often decelerate sharply immediately after reaching the peak forward velocity, indicating inefficient energy use.

In the middle portion of each operation following the initial acceleration, the OTF controller tends to cruise while gradually pitching backwards and decelerating slightly, whereas the baseline controllers maintain high pitch angle and forward acceleration later into the operation. In this way, the peak forward velocities of the optimized trajectories avoid exceeding the energy-optimal forward cruise velocity shown in Fig. 6. This behavior also allows for less extreme pitch angles at high forward velocities, improving energy efficiency by increasing the lift-to-drag ratio as discussed in Sec. 2.6. The baseline controllers instead maintain a steeper pitch angle, causing increased power requirements over the middle portion of the operation. This effect can be clearly observed from roughly 4 to 7.5 $s$ in Fig. 8(e), during which time the total energy used by the baseline controllers, especially HVB, is shown to increase much more rapidly than the energy used by the OTF controller.

The optimized deceleration behaviors also demonstrate significant energy saving over the those of the baseline controller. The improvements are in part related to the previous cruise portion of the operation, during which the OTF controller allows the vehicle to start deceleration gradually. Consequently, body drag supplements active braking, reducing both the time and energy costs of the final deceleration portion, compared to the baseline controllers which must actively brake from a higher velocity. Additionally, the optimized trajectories are able to brake more precisely by considering the full vehicle dynamics. For the baseline controllers, the target velocity is directly proportional to the remaining distance to the endpoint. This approach causes the vehicle to gradually slow down as it approaches the endpoint, requiring significant time spent at low and energy-inefficient velocities. As a result, the baseline controllers are less precise than the OTF controller optimized based on the full vehicle dynamics, which allows the vehicle to stop more promptly and efficiently.

Across all flight segments, the OTF controller is also able to balance the forward and horizontal motion of the vehicle more efficiently than the baseline controllers. An example can be observed in the second column of Fig. 8, i.e., subplots (f)–(j), showing the trajectories for a diagonal flight with simultaneous horizontal and vertical motion. In subplot (h), it is seen that the optimized trajectory reaches and maintains the steepest forward pitch angle $Θ$ early in the motion from approximately 0.5 to 2 $s$⁠, during which time the vertical position $Z$ increases slowly as shown in subplot (i). Meanwhile, the baseline controllers are at their steepest forward pitch angles from approximately 2.5 to 4.5 $s$⁠, when $Z˙$ is also around its peak. According to the previous analysis in Sec. 2.6, high pitch angle, when coupled with large perpendicular inflow velocity $vz$ (which increases with vehicle vertical velocity $Z˙$⁠), would decrease the rotor energy efficiency by reducing the lift-to-drag ratio. Consequently, the baseline controllers will suffer lower energy efficiency compared to the OTF controller, due to the coupling of high pitch angle with high vertical velocity during the course of the motion. The effects of these behaviors can be observed in subplot (j), which shows that, although the power used by the OTF controller is initially higher than that of either baseline controller, the latter increased much faster during the middle portion of the operation, and the total energy used is roughly equal at around 5.5 $s$⁠. At that point, according to subplots (f) and (i), the OTF controller has almost reached the endpoint, while the baseline controllers still have significant forward and upward distance to cover. Specifically, at this time, the vehicle is 3.8 $m$ from the endpoint (3.6 $m$ horizontal, 1.2 $m$ vertical) under the OTF controller, compared to 12.6 $m$ (11.9 $m$ horizontal, 4.2 $m$ vertical) under the HVB controller. The comparison testifies the advantage of the OTF controller in energy efficiency (per distance).

It should also be noted that the calibration and constraint settings generally need to be more conservative for the baseline controllers than for the optimized trajectory-following controller, reducing the operation range as well as the potential of energy saving. This is because the achievable operating limits of the UAV, governed by the underlying fundamental physics, could change under different vehicle conditions. For example, as shown in Ref. [32], battery energy depletion over the course of a flight can cause significant reduction in the maximum attainable thrust by the propeller, due to the impacts of battery voltage on propulsion through the motor and ESC dynamics. Therefore, assuming that the vehicle is appropriately modeled, the optimized trajectories could exploit the full energy saving potential while following the (varying) limits accurately by taking into account the vehicle physical dynamics. By contrast, it would be difficult for the baseline controllers to accommodate both aspects of the performance. On one hand, if the controller is calibrated under the full battery condition, it is likely to request unattainable thrust under low battery, e.g., near the end of the flight, potentially resulting in unsafe operating conditions. On the other, if the controller parameters and constraints are set conservatively to accommodate all operating conditions, the energy performance would be sacrificed.

Finally, based on Table 1, the performance of the low-velocity baseline controller can also be evaluated. While this controller does generally have improved energy efficiency over the high-velocity baseline controller, it is only able to achieve a small fraction of the improvement by the optimized trajectory following controller. For example, for the 100 $m$ horizontal flight, the LVB controller reduces the energy consumption over HVB by 2.8%, while the OTF controller achieves a much higher reduction of 9.2%. It is seen from Fig. 8(b) that the improvement by the LVB controller is mainly achieved by keeping the vehicle at the energy-optimal horizontal velocity (which is 12.6 $m/s$ as identified from Fig. 6) during the middle cruising portion of the operation, while the velocity under the HVB controller is higher and less energy efficient. Such improvement is more significant in flights with a longer horizontal component as seen across Table 1. Nevertheless, the minimal improvement in comparison with the OTF controller demonstrates the importance of optimizing not only the steady-state behavior of the UAV, e.g., cruising velocity, but also the dynamic maneuver, e.g., acceleration, deceleration, and pitching, based on the holistic vehicle system-level dynamics.

In this section, experimental testing results are presented to validate the performance of the optimized trajectories and feedback controllers. The testing platform is described first in this section, followed by the experimental procedures and result analysis.

A test vehicle, shown in Fig. 9, has been built by using the airframe and propulsion system (including propellers, motors, and ESC) of a DJI Spreading Wings S1000+ octorotor, and retrofitting the control and sensing systems with a Pixhawk 2.1 Flight Controller, a Piksi Multi-GNSS receiver, a 3DR uBlox GPS with compass, a battery current transducer, and a voltage sensor. With this setup, the multirotor can be controlled to perform different flight operations with data measured and recorded for analysis and validation. Regarding the control software, as described in Sec. 3.2, the feedback controllers of the developed framework are implemented using the custom script of the ArduPilot Mission Planner in a two-layer PID control architecture. The high-level control runs in a ground station computer, which communicates with the low-level control in the onboard flight controller for exchange of command and sensor information as well as data logging. Each of the vehicle states required for feedback control as shown in Fig. 7 are calculated from a combination of measurements from the vehicle's GPS, accelerometer, and gyroscope hardware. The sensor measurements are processed and filtered by the onboard flight controller, providing accurate measurements of the required vehicle rigid-body states which can be accessed in real-time for feedback control. As the benchmark for comparison, the aforementioned two baseline controllers are also implemented using the waypoint-based Auto-Mode navigation function of the Mission Planner.

The total thrust $ΣTj$⁠, and the magnitude and direction of its horizontal component $ΣTj,horiz$⁠, can be calculated using the equations in Sec. 2.4 and the measured orientation of the vehicle. Using this approach, the wind was measured multiple times over the course of testing to account for the changing wind conditions. For each measurement, the trim condition was recorded at multiple yaw angles to improve accuracy and compensate for potential sensor bias. The overall average wind speed was found to be approximately 5 m/s heading South at the day of testing, which agreed with weather reports. Specific wind measurements for each operation are provided in Sec. 4.3.

The experimental testing results are presented in Table 2. This table summarizes the total energy consumption under each controller for each operation and vehicle heading, as well as the improvement over the HVB controller. The wind conditions for each group of operations are also specified, including wind velocity and direction (vehicle heading) measured by angle clockwise from North. It is noted that, due to time and battery energy constraint during testing, results for the LVB controller are only available for 50 m and 100 m forward flights. As mentioned in Sec. 3.3, to consistently compare each controller, the cumulative energy consumption for each operation was calculated at the time when the vehicle entered a certain radius around the endpoint (increased to 4 $m$ to accommodate greater variance due to wind disturbances and sensor noise under experimental conditions).

Based on these results, the developed trajectory optimization and feedback control framework is shown to achieve a significant reduction in energy use relative to both baseline controllers. Averaged across all operations, the proposed OTF controller based on the optimized trajectories uses 10.2$%$ less energy than the high-velocity baseline controller, and 8.3% less than the low-velocity baseline. The observed improvements are in good match with those from simulation testing in Sec. 3.3, which show average improvement of 10.7% and 9.0%, respectively. To examine individual operations in more detail, Fig. 10 shows the trajectories of key vehicle states for 50$m$ forward flights under different headings using the OTF and HVB controllers. These experimental results demonstrate that the energy performance improvements achieved in simulation are largely captured in real-world operation, and that the overall behaviors of each controller are similar. Meanwhile, the improvements achieved by the OTF controller are slightly worse than those in simulation due to the uncertainties and disturbances under real-world experimental conditions. In particular, wind has been shown to significantly affect the vehicle, and several specific behaviors are observed, which explain the discrepancy between simulated and real-world performance.

One behavior beneficial to the energy performance of the OTF controller relative to the baseline controllers is the larger variation of the forward velocity with respect to wind. As described in Sec. 3.2, the baseline control architecture sets the pitch angle command to follow a forward velocity target, which is dependent on the remaining distance to the endpoint. Consequently, under the baseline controller, the only rigid-body state with significant variation is the pitch angle, allowing the vehicle to track the forward velocity target closely under whatever wind conditions. By contrast, the OTF controller calculates the pitch angle command using the errors of three states relative to the target trajectories, i.e., forward position, forward velocity, and pitch angle, each with an assigned weight. Under wind disturbance, the controller would not be able to track all three states exactly, and this effect actually helps improve the energy efficiency over the baseline controllers. Specifically, the maximum forward velocity reached by the OTF controller is generally lower when traveling against the wind, and higher when traveling with the wind. This variation, which can be observed in the OTF forward position trajectories shown in Fig. 10(a), coincides with the trend of the energy-optimal cruising velocity with respect to wind, as shown in Fig. 6. Consequently, the vehicle velocity under the OTF controller remains closer to the wind-adjusted optimal forward velocity near the middle segment of the operation, while the baseline controllers reach approximately the same maximum forward velocity regardless of wind. This difference is particularly notable due to the high forward velocity target of the HVB controller, at which the energy cost per meter can increase significantly with headwind. As a result, despite the consistency of the forward position and velocity profiles over different wind conditions, the power used by the baseline controller, shown in Fig. 10(f), is observed to increase considerably near the middle segment of the operation when traveling against the wind.

Meanwhile, there are some other behaviors of the OTF controller that could lead to degraded energy performance under real-world experimental conditions. First, the OTF controller tends to be more sensitive to disturbances, as demonstrated in Fig. 10 by the larger variation of the vehicle state trajectories compared to those of the baseline controller. This is in part due to the PID gains of the OTF controller, which are calibrated to be more aggressive and responsive compared to the baseline controllers, enabled by the underlying physical model. While the responsiveness improves the energy performance as discussed in Sec. 3.3, it would also make the OTF controller more sensitive to wind disturbances. Second, the vehicle trajectories are currently optimized as time sequences, which also makes trajectory following more susceptible to wind disturbances. For example, the braking portion of the trajectory occurs at a fixed time regardless of the position of the vehicle, and hence, the vehicle may sometimes slow down earlier than expected when traveling against the wind. This would result in undershoot in trajectory following, which requires the vehicle to accelerate again after braking, leading to waste of energy. This effect tends to be more significant in longer flights, due to the accumulation of position errors caused by wind disturbances. While this effect does not cause substantial degradation in energy efficiency, the vehicle energy performance can be further improved by parameterizing the trajectory in terms of vehicle states, e.g., forward position and velocity, instead of time. One approach is to formulate trajectory optimization as an optimal control problem and derive control policies, which could perform trajectory generation in real-time based on feedback of the actual vehicle states.

It is also noted that there were some fluctuations in wind conditions across flights that could affect the comparisons. For instance, in the 100 m forward flight case, the wind speed was higher for operations under the OTF controller (5.3 $m/s$⁠) than those under the HVB controller (3.9 $m/s$⁠). This had a particularly significant impact on the performance of the northbound operations (i.e., against the wind), where the baseline slightly outperformed the OTF controller, in part due to the wind speed favoring the former. Although the opposite effect occurs when traveling with the wind, the impact is less substantial, as it can be observed in Fig. 6 that the effect of wind opposite vehicle motion is generally greater than that of the same wind assisting vehicle motion. In the 70 $m$ forward flight case, the performance improvement of the OTF controller (over HVB) is instead the best when traveling North and worst when traveling South, while the opposite is true in the 100 $m$ case, as wind speed during the OTF controller operation is lower than that during the HVB operation in the 70 m case but higher in the 100 m case. This effect does not generally favor either the HVB or OTF controllers, but introduces uncertainty and fluctuations in the results, which is the main reason why experiments were repeated multiple times in various directions to ensure more fair comparison of controller performance.

The OTF controller achieves the largest improvement over the baseline controllers in the diagonal flight case (⁠$Xf=50$⁠, $Zf=20$⁠) in experiments (−19.8% on average), which is significantly higher than in simulation (−9.3%). In both cases, the improvement is largely due to the ability of the optimized trajectories to balance both horizontal and vertical motion efficiently, as described in Sec. 3.3. However, the more prominent performance improvement observed in the experimental data is also caused by a baseline controller behavior not seen in simulation. Specifically, the HVB controller was observed to decelerate horizontally earlier than expected in diagonal flight, requiring the vehicle to take additional time and energy regaining forward velocity to reach the endpoint. This behavior appears to be a result of the real-world implementation of the controller, which tries to avoid overshoot by attempting to reach the vertical and horizontal end states simultaneously. This issue demonstrates the difficulty of calibrating a robust and efficient baseline controller capable of balancing motion along multiple axes, due to the total reliance on feedback with no knowledge of underlying system physics.

Finally, it can be observed that the energy consumption for each controller and operation group in experiments is generally lower than that in simulation. One possible cause is the larger radius about the target point used in experimental testing to identify the end of operation, which was necessary to accommodate the greater variance introduced by wind effects. As another possible explanation, there may be slight discrepancies between the UAV model and the real-world test vehicle. For example, the body drag coefficient was identified using data from previous experiments, by fitting the model prediction of forward acceleration to measurement under horizontal vehicle motion. Although multiple trials were used to minimize variance, there are potential uncertainties, such as those introduced by wind effects, which would induce errors in the result. However, despite these discrepancies, the OTF controller achieves significant reduction in energy consumption in all cases of flights, and the improvement over the baseline controllers is similar in experiments and simulation, demonstrating the effectiveness of the developed planning and control framework to improve vehicle energy performance in real-world applications.

In this paper, an energy-optimal multirotor trajectory planning and feedback control framework has been developed based on a system model, which includes all major subsystem physical dynamics. In simulation, the framework was found to achieve an average energy saving of 10.7$%$ over a baseline controller, which takes a conventional PID feedback control architecture. The developed framework reduces the energy consumption by holistically planning vehicle behavior over the course of a flight and coordinating multiple vehicle states and motion along different coordinates. Several behaviors contributing to performance improvements were described based on a comparison of the baseline controller and the proposed framework in two sample operations. For example, the framework enables the vehicle to supplement active braking with passive deceleration from air resistance, cruise around energy-optimal horizontal velocity, and accelerate more rapidly and efficiently. Additionally, a more efficient version of the baseline controller was also tested, which uses the energy-optimal velocity derived from the model as the velocity limit. This controller was only able to achieve a small (1.9$%$⁠) average reduction in energy performance over the original baseline, showing the importance of optimizing not only the steady-state vehicle behavior, e.g., cruising velocity, but also the dynamic maneuver. Finally, the framework was validated in real-world vehicle testing and shown to capture most of the improvements observed in simulation with a 10.2$%$ average reduction in energy cost relative to the baseline controller, although disturbances under experimental conditions, e.g., wind effects, cause certain degree of variances and fluctuation. Both experimental and simulation results demonstrate the significant energy-efficiency improvements enabled through model-based planning and control by considering and coordinating the integrated system physical dynamics.

The sponsor was not directly involved in study design, collection, analysis and interpretation of data, writing of this article, or decision making to submit this article for publication.

• Office of Naval Research (Award ID: N00014-21-1-2080; Funder ID: 10.13039/100000006).