Multirotor aerial vehicle with automatically rotatable arms

Abstract

A multirotor aerial vehicle (MAV) is disclosed. The MAV includes a housing, a plurality of rotatable arms, wherein each of the plurality of rotatable arms has a proximal end coupled to the housing and a distal end configured to rotate about a vertical axis passing through the proximal end of the corresponding arm, a plurality of thrust-generating rotors, each coupled to a corresponding one of the plurality of rotatable arms at the corresponding distal end, a flight controller configured to selectively control each of the plurality of thrust-generating rotors, and a flight trim controller configured to control rotation of the plurality of rotatable arms in order to adjust the geometric center of the rotors of the MAV from a first center of gravity (CoG) associated with the MAV in an unloaded state to a second CoG associated with the MAV in a loaded state.

Claims

1. A multirotor aerial vehicle (MAV), comprising: a housing; a plurality of arms, each of the plurality of arms having a proximal end coupled to the housing and a distal end, and at least one is configured to rotate about a vertical axis passing through the proximal end of the corresponding arm; a plurality of thrust-generating rotors, each coupled to a corresponding one of the plurality of arms at the corresponding distal end; a flight controller configured to selectively control each of the plurality of thrust-generating rotors; and a flight trim controller configured to control rotation of at least one of the plurality of arms in order to adjust the geometric center of the rotors of the MAV from a first center of gravity (CoG) associated with the MAV in an unloaded state to a second CoG associated with the MAV in a loaded state, wherein each of the plurality of arms is rotatable, and where angular positions of the plurality of arms are ζ.sub.1, ζ.sub.2, ζ.sub.3, and ζ.sub.4 when the plurality of arms includes four arms capable of rotation, where ζ.sub.1, ζ.sub.2, ζ.sub.3, and ζ.sub.4 are calculated by determining a bias torque (τ.sub.g) as a torque resulting from shifting of the first CoG to the second CoG, where τ.sub.g is defined as ${\tau }_g=l_g\times F_g\mathrm{or}\left[\begin{array}{c}{\tau }_{\mathrm{gx}}\\ {\tau }_{\mathrm{gy}}\\ {\tau }_{\mathrm{gz}}\end{array}\right]=\left[\begin{array}{c}{l}_y{F}_z-l_z{F}_y\\ l_z{F}_x-l_x{F}_z\\ l_x{F}_y-l_y{F}_x\end{array}\right],$ where l.sub.g represents the location of the CoG defined by [l.sub.x l.sub.y l.sub.z].sup.T, and gravity of the MAV is F.sub.g defined by [F.sub.x F.sub.y F.sub.z].sup.T which has a magnitude mg where m is the total mass of the MAV plus a payload and g is the gravitational constant, where rotor thrust F.sub.i (i=1, 2, 3, 4), the magnitude of the total thrust F.sub.r, and the total torque generated by all rotor thrust τ.sub.r, are governed by: $\left[\begin{array}{c}{F}_1\\ F_2\\ F_3\\ F_4\end{array}\right]=R\left[\begin{array}{c}{\tau }_{\mathrm{rx}}\\ {\tau }_{\mathrm{ry}}\\ {\tau }_{\mathrm{rz}}\\ F_r\end{array}\right]$ where R is a rotation matrix defined as $R={\left[\begin{array}{cccc}a\sin {\zeta }_1& a\sin {\zeta }_2& a\sin {\zeta }_3& a\sin {\zeta }_4\\ a\cos {\zeta }_1& a\cos {\zeta }_2& a\cos {\zeta }_3& a\cos {\zeta }_4\\ -\frac{{\tau }_{d1}}{F_1}& -\frac{{\tau }_{d2}}{F_2}& -\frac{{\tau }_{d3}}{F_3}& -\frac{{\tau }_{d4}}{F_4}\\ 1& 1& 1& 1\end{array}\right]}^{-1}$ where τ.sub.di (i=1, 2, 3, 4) represents the aerodynamic drag of the ith rotor, and τ.sub.di is proportional to thrust of the ith rotor (F.sub.i), angular positions of all four arms (ζ.sub.i, ζ.sub.2, ζ.sub.3, and ζ.sub.4) can be solved by satisfying: $\hspace{1em}\{\begin{array}{c}{\tau }_{\mathrm{gx}}+a\underset{i=1}{\overset{4}{.Math.}}\sin {\zeta }_i{F}_i=0\\ {\tau }_{\mathrm{gy}}+a\underset{i=1}{\overset{4}{.Math.}}\cos {\zeta }_i{F}_i=0\end{array}$ with an objective function being minimized.

2. The MAV of claim 1, wherein the at least one of the plurality of arms capable of rotating is actuated by an actuator.

3. The MAV of claim 2, wherein the actuator is a servo.

4. The MAV of claim 1, wherein each of the plurality of arms is actuated by an actuator.

5. The MAV of claim 4, wherein each of the actuators is a servo.

6. The MAV of claim 1, further comprising an attitude sensor to determine attitude parameters of the MAV, including roll, yaw, and pitch.

7. The MAV of claim 1, wherein the objective function is ƒ(ζ.sub.n)=Σ.sub.i=1.sup.4|ζ.sub.i-ζ.sub.ƒi|, where ζ.sub.ƒi (i=1, 2, 3, 4) represents the arm-associated angle of the ith arm of a conventional QR whose arms are fixed at [ζ.sub.ƒ1 ζ.sub.ƒ2 ζ.sub.ƒ3 ζ.sub.ƒ4].sup.T=[45° 135°-135°-45°].sup.T.

8. The MAV of claim 1, wherein a constant of proportionality between τ.sub.di and F.sub.i is between 0.018 and 0.022.

9. A method of stabilizing a multirotor aerial vehicle (MAV), comprising: receiving a payload to be transported by the MAV; determining the bias torque caused by the shift of the CoG due to the eccentric payload to be transported by the MAV; determining the target positions of at least one of the plurality of arms capable of rotating with respect to a vertical axis passing through a point near a proximal end of the arm; rotating the at least one of the plurality of arms capable of rotating in order to adjust the geometric center of the rotors of the MAV from a first center of gravity (CoG) associated with the MAV in an unloaded state to a second CoG associated with the MAV in a loaded state; applying thrusts to a plurality of rotors, each positioned at a distal end of a corresponding arm, wherein each of the plurality of arms is rotatable, and where angular positions of the plurality of arms are ζ.sub.1, ζ.sub.2, ζ.sub.3, and ζ.sub.4 when the plurality of arms includes four arms capable of rotation, where ζ.sub.1, ζ.sub.2, ζ.sub.3, and ζ.sub.4 are calculated by determining a bias torque (τ.sub.g) as a torque resulting from shifting of the first CoG to the second CoG, where τ.sub.g is defined as ${\tau }_g=l_g\times F_g\mathrm{or}\left[\begin{array}{c}{\tau }_{\mathrm{gx}}\\ {\tau }_{\mathrm{gy}}\\ {\tau }_{\mathrm{gz}}\end{array}\right]=\left[\begin{array}{c}{l}_y{F}_z-l_z{F}_y\\ l_z{F}_x-l_x{F}_z\\ l_x{F}_y-l_y{F}_x\end{array}\right],$ where l.sub.g represents the location of the CoG defined by [l.sub.x l.sub.y l.sub.z].sup.T, and gravity of the MAV is F.sub.g defined by [F.sub.x F.sub.y F.sub.z].sup.T which has a magnitude mg where m is the total mass of the MAV plus a payload and g is the gravitational constant, where rotor thrust F.sub.i (i=1, 2, 3, 4), the magnitude of the total thrust F.sub.r, and the total torque generated by all rotor thrust τ.sub.r, denoted by τ.sub.r=[τ.sub.rx τ.sub.ry τ.sub.rz].sup.T, have the following relationship: $\left[\begin{array}{c}{F}_1\\ F_2\\ F_3\\ F_4\end{array}\right]=R\left[\begin{array}{c}{\tau }_{\mathrm{rx}}\\ {\tau }_{\mathrm{ry}}\\ {\tau }_{\mathrm{rz}}\\ F_r\end{array}\right]$ where R is a rotation matrix defined as $R={\left[\begin{array}{cccc}a\sin {\zeta }_1& a\sin {\zeta }_2& a\sin {\zeta }_3& a\sin {\zeta }_4\\ a\cos {\zeta }_1& a\cos {\zeta }_2& a\cos {\zeta }_3& a\cos {\zeta }_4\\ -\frac{{\tau }_{d1}}{F_1}& -\frac{{\tau }_{d2}}{F_2}& -\frac{{\tau }_{d3}}{F_3}& -\frac{{\tau }_{d4}}{F_4}\\ 1& 1& 1& 1\end{array}\right]}^{-1}$ where τ.sub.di (i=1, 2, 3, 4) represents the aerodynamic drag of the ith rotor, and τ.sub.di is proportional to thrust of the ith rotor (F.sub.i), angular positions of all four arms (ζ.sub.i, ζ.sub.2, ζ.sub.3, and ζ.sub.4) can be solved by satisfying: $\hspace{1em}\{\begin{array}{c}{\tau }_{\mathrm{gx}}+a\underset{i=1}{\overset{4}{.Math.}}\sin {\zeta }_i{F}_i=0\\ {\tau }_{\mathrm{gy}}+a\underset{i=1}{\overset{4}{.Math.}}\cos {\zeta }_i{F}_i=0\end{array}$ with an objective function being minimized.

10. The method of claim 9, wherein the at least one of the plurality of arms capable of rotating is actuated by an actuator.

11. The method of claim 10, wherein the actuator is a servo.

12. The method of claim 9, wherein each of the plurality of arms is actuated by an actuator.

13. The method of claim 12, wherein each of the actuators is a servo.

14. The method of claim 9, further comprising determining attitude parameters of the MAV, including roll, yaw, and pitch.

15. The method of claim 9, wherein the objective function is ƒ(ζ.sub.n)=Σ.sub.i=1.sup.4|ζ.sub.i-ζ.sub.ƒ1|, where ζ.sub.ƒi (i=1, 2, 3, 4) represents the arm-associated angle of the ith arm of a conventional QR whose arms are fixed at [ζ.sub.ƒ1 ζ.sub.ƒ2 ζ.sub.ƒ3 ζ.sub.ƒ4].sup.T=[45° 135°-135°-45° ].sup.T.

16. The method of claim 9, wherein a constant of proportionality between τ.sub.di and F.sub.i is between 0.018 and 0.022.

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) FIG. 1 is a schematic of a control scheme, depicting blocks in a system including a flight trim controller and a flight controller capable of adjusting the geometric center of the rotors of a multirotor unmanned aerial vehicle, e.g., a quadrotor (QR), according to one embodiment of the present disclosure.

(2) FIG. 2 is a schematic representation of a QR in an unloaded state (i.e., with no attached payload).

(3) FIG. 3 is a schematic representation of the QR of FIG. 2 in a loaded state (i.e., with an attached payload).

(4) FIG. 4 is an action flowchart provided to explain the actions of the flight trim controller and the flight controller of FIG. 1.

DETAILED DESCRIPTION

(5) For the purposes of promoting an understanding of the principles of the present disclosure, reference will now be made to the embodiments illustrated in the drawings, and specific language will be used to describe the same. It will nevertheless be understood that no limitation of the scope of this disclosure is thereby intended.

(6) In the present disclosure, the term “about” can allow for a degree of variability in a value or range, for example, within 10%, within 5%, or within 1% of a stated value or of a stated limit of a range.

(7) In the present disclosure, the term “substantially” can allow for a degree of variability in a value or range, for example, within 90%, within 95%, or within 99% of a stated value or of a stated limit of a range.

(8) A novel approach is provided in the present disclosure to compensate for a shift in center of gravity (CoG) or an associated bias torque of a quadrotor (QR) that does not interfere with the QR's ability to manage demands for varying the thrust generated by each rotor. For generality, unmanned aerial vehicles (UAVs) are referred to as QRs or multirotor aerial vehicles (MAVs), interchangeably. However, as discussed more fully below, it is understood that more or less number of rotors than four are possible and are within the scope of this disclosure.

(9) Referring to FIG. 1, a system 10 is provided, depicting various blocks according to the present disclosure capable of adjusting the geometric center of the rotors 152.sub.i (where i is the subscript representing rotors of the physical system, e.g., rotors 152.sub.1, 152.sub.2, 152.sub.3, and 152.sub.4) of a QR 100 or 100′ (shown in FIGS. 2 and 3) with an insignificant impact on thrust. The system 10 includes a Flight Controller 60 adaptable to provide general flight controls for the QR 100 or 100′ (see FIGS. 2 and 3) and a Flight Trim controller 61 adaptable to provide needed controls to trim the QR 100 or 100′ (see FIGS. 2 and 3). The Flight Controller 60 receives data from a block 63 identified as “Flight Data” which provides destination information, such as latitude and longitude, of the QR 100 or 100′ (see FIGS. 2 and 3). This information may be obtained manually or provided as an external input as provided by the block 67. The Flight Trim controller 61 receives its input, such as roll, yaw, and pitch data, from one or more attitude sensors 66.

(10) As provided above, the system 10 includes two controllers: 1) Flight Controller 60, and 2) Flight Trim Controller 61. While two controllers are shown, a single controller can be implemented responsible for functionalities, described herein, of each of the shown controllers. Additionally, the Flight Trim Controller 61 can be maintained separately as part of a retrofit assembly to an existing Flight Controller 60 in a conventional QR, with minimal electronic disturbances.

(11) As mentioned above, the Flight Controller 60 receives flight data from the block 63 from a subsystem based on manual input or external data as provided in block 67. The flight data of block 63 can include origination coordinates, destination coordinates, GPS data, real-time attitude and speed requirements, and other parameters known to a person having ordinary skill in the art. The Flight Controller 60 processes the flight data of block 63 and converts that data into thrust information for each of the rotors as shown in blocks 52.sub.i (where i is the subscript representing rotors of the physical system, e.g., rotors 152.sub.1, 152.sub.2, 152.sub.3, and 152.sub.4 of the QR 100 or 100′ shown in FIGS. 2 and 3) which is sent to the Flight Trim Controller 61. Depending on the rotor design, the thrust information can take the form of a digital signal or an analog signal. These rotor blocks are shown as 52.sub.1 to 52.sub.n representing rotors 1 through n. As mentioned above, n can be any integer from two to eight.

(12) At the same time, one or more attitude sensors 66 provide attitude data including roll, yaw, and pitch to the Flight Trim Controller 61. Alternatively, the attitude data from the sensors can be received by the Flight Controller 60 and communicated to the Flight Trim Controller 61. The Flight Trim Controller 61 receives the attitude data from the one or more attitude sensors 66 as well as the rotor thrust information from the Flight Controller 60. The latter can be in the form of a bus or individual I/O communication lines between the Flight Controller 60 and Flight Trim Controller 61.

(13) Each rotor control block 52 is coupled to a rotatable arm control block 42 (i.e., arm 1 to arm n). Each rotatable arm control block 42 is selectively configured to be rotated as discussed in greater detail below. Initially with the QR 100′ (see FIG. 2) in an unloaded state (i.e., no payload attached to the QR), the QR's CoG 171′ (see FIG. 2) may coincide with its geometric center 181′ (see FIG. 2). However, the QR 100 (see FIG. 3) in a loaded state (i.e., based on the attachment of a payload 148 (see FIG. 3)) has a new geometric center 181 and a new CoG 171 (see FIG. 3). The new geometric center 181 is the geometric center of the planar geometry formed by the locations of the four rotors 152.sub.i of the QR 100. As the rotating arms 142.sub.i rotates, the rotors 152.sub.i move and the position of the new geometric center 181 changes accordingly. The geometric center 181 is located in a certain area of the planar geometry formed by the locations of the four rotors 152.sub.i of the QR 100. The original CoG 171′ is moved to the new CoG 171 due to the acquirement of the payload 148. According to the present disclosure, the geometric center 181 should be moved to the location above the CoG 171 (see FIG. 3) by rotating the four rotating arms 142.sub.i (where the subscript i represents each arm, i.e., in the case of FIG. 3, i=1, 2, 3, and 4). The line connecting the geometric center 181 and the CoG 171 (see FIG. 3) is parallel to the z axis. To that end, the Flight Trim Controller 61 can bring the geometric center 181 to the location where the CoG 171 (see FIG. 3) is projected onto the plane formed by the rotors 152.sub.i of the QR 100 in a steady attitude by selectively rotating each of the rotatable arms 142.sub.i (see FIGS. 2 and 3, where the subscript i represents 1-4). The Flight Trim Controller 61 selectively controls the position of each rotatable arm 142.sub.i (see FIGS. 2 and 3) by controlling a corresponding actuator control block 46 which may be a system of servos (e.g., servo 1 through servo n). However, actuation is not limited to the use of servos wherein other actuation mechanisms known to a person having ordinary skill in the art are also possible. It is possible to have a smaller number of actuators than rotatable arms, by providing the appropriate linkage, known to a person having ordinary skill in the art. The signal received by the actuator control blocks 46.sub.1 through 46.sub.n from the Flight Trim Controller 61 may be in the form of a digital signal or analog signal depending on the design of the actuators.

(14) The Flight Trim Controller 61 provides signals indicative of arm position for each of the rotatable arms 142.sub.i (see FIGS. 2 and 3) to the Flight Controller 60. This can be in the form of a bus or individual I/O communication lines between the Flight Controller 60 and Flight Trim Controller 61.

(15) Optionally, each rotor control block 52.sub.i (i.e., for rotor 1 to rotor n) may be equipped with a force sensor 64 whose data (i.e., F.sub.1 to F.sub.n) can be correlated to a thrust signal in communication with the Flight Trim Controller 61. These signals can be, in addition to the corresponding signals, received from the Flight Controller 60, in which case the Flight Controller 60 and Flight Trim Controller 61 may go through an initial calibration at the beginning of each power cycle, or these signals may replace the thrust information provided by the Flight Controller 60. Additionally, each rotatable arm 42 (i.e., arm 1 to arm n) can be equipped with a position sensor 65.sub.i (e.g., an optical encoder) that can provide position data for the respective rotatable arm 142.sub.i (see FIGS. 2 and 3), i.e., ζ.sub.1 to ζ.sub.n as a feedback signal to the Flight Trim Controller 61. The Flight Trim Controller 61 may be configured to operate in an open-loop configuration without the position sensors 65.sub.i, or in a closed-loop manner using the position sensor data obtained from the position sensors 65.sub.i.

(16) In the present disclosure, QRs 100 with rotatable arms are referred to as QRAs and QRs 100 with fixed arms are referred to as QFAs. Therefore, while not shown, it is possible that some of the arms are rotatable while some of the arms are fixed. Each of the rotatable arms in a QRA can selectively rotate in the x-y plane of F.sub.B within a certain range while the arms of a QFA are fixed. It should be appreciated that rotor thrust of each rotor 152.sub.i (see FIGS. 2 and 3) of a QRA is normal to the x-y plane of F.sub.B which is described in more detail below.

(17) Referring to FIGS. 2 and 3, perspective views of the QR 100 (FIG. 2 representing a QR in an unloaded state, i.e., without a payload) and QR 100′ (FIG. 3 representing the QR in a loaded state, i.e., with a payload attached) are shown, respectively. The non-limiting embodiment of the QR 100 and 100′ as shown in FIGS. 2 and 3 includes four rotatable arms 142.sub.i (142.sub.1, 142.sub.2, 142.sub.3, and 142.sub.4), with each starting from a housing 151 and having at its distal end (144.sub.1, 144.sub.2, 144.sub.3, and 144.sub.4) a corresponding rotor 152.sub.i (152.sub.1, 152.sub.2, 152.sub.3, and 152.sub.4) rotating corresponding propellers 153.sub.i (153.sub.1, 153.sub.2, 153.sub.3, and 153.sub.4). The housing 151 shown in FIG. 2 is depicted as, but not limited to, having a cylindrical structure. Rotation of the rotatable arms 142.sub.i is achieved through use of an actuator 46 (referenced in FIG. 1), which is controlled by the Flight Trim Controller 61. The actuator blocks 46.sub.i (see FIG. 1) used may be, but not limited to, a system of servos wherein other actuation mechanisms known to a person having ordinary skill in the art are also possible, as described above. Each rotor 152.sub.i generates a normal thrust indicated by F.sub.1, F.sub.2, F.sub.3, and F.sub.4. Adjustment of the geometric center of the rotors 152.sub.i is accomplished by the rotation of corresponding rotatable arms 142.sub.i, each with associated rotors 152.sub.i. The rotation of the rotatable arms 142.sub.i occurs, as discussed above, by the combination of blocks 52.sub.1 (see FIG. 1, titled Rotor 1 which includes force sensor 64.sub.1), 42.sub.1 (titled Arm 1 which includes position sensor 651), and 46.sub.1 (titled Actuator 1) for a rotatable arm 142.sub.1 (see FIGS. 2 and 3) . . . and the combination of blocks 52.sub.n (titled Rotor n which includes force sensor 64.sub.n), 42.sub.n (titled Arm n which includes position sensor 65.sub.n), and 46.sub.n (titled Actuator n) for a rotatable arm 142.sub.n (see FIGS. 2 and 3, where in the case of FIG. 2 n=4). As discussed above, the rotatable arms 142.sub.i are rotated about the housing by angles ζ.sub.1, ζ.sub.2, ζ.sub.3, and ζ.sub.4, respectively, by use of actuators (e.g., servo motors) disposed in the housing 151.

(18) As further discussed above, an important goal of the novel arrangement disclosed herein is to adjust the geometric center of the rotors 152.sub.i of the QR 100 to coincide with a shift in the location of the CoG 171 of the QR 100 upon acquiring a payload 148. Any shift of the CoG 171 from the existing geometric center of the rotors 152.sub.i of the QR 100 results in a CoG offset. For example, a CoG offset can occur when a first CoG associated with the QR 100 in an unloaded state, is shifted to a second CoG associated with the QR 100 in a loaded state (e.g. when a package or payload 148 is loaded). When the geometric center of the rotors 152.sub.i of the QR 100 is changed to coincide to the projection of the CoG 171 on the plane formed by the rotors 152.sub.i, the CoG offset is zero and thus, there is no bias torque caused by the CoG offset. While a shift in the CoG can create an associated bias torque, other external factors (e.g., a steady prevailing wind, or a sudden gust) can also create a bias torque. The novel approach presented here is adaptable to account for bias torque resulting from a shifting CoG as well as those which are caused by the aforementioned external factors.

(19) Thus, a novel approach is disclosed to compensate for a shift in the CoG 171 or a change in bias torque resulting from the shift in CoG or from external factors. This novel approach does not interfere with the ability of the QR 100 to manage demands for varying the thrust generated by rotors 152.sub.i of the QR 100 (see FIGS. 2 and 3). While the present disclosure is derived based on unmanned aerial vehicles (UAVs) with four rotors 152.sub.1, 152.sub.2, 152.sub.3, and 152.sub.4 (see FIGS. 2 and 3), it should be appreciated that the concepts discussed herein are applicable to, and thus within the scope of the present disclosure, UAVs with more or less than four rotors (e.g., as few as two rotors and as many as eight rotors) are within the scope of the present disclosure.

(20) A non-zero CoG offset is just one of many sources that can contribute to the bias torque. Other sources can be a manipulation force (when a robot manipulator (not shown) installed on the QR 100 interacts with the environment), aerodynamic drag (caused by the rotation of propellers 153.sub.i, and so on. Propellers 153.sub.i depicted in FIG. 2, are of a common dual-blade design, however the present disclosure is not limited to the use of such propellers 153.sub.i where alternate propeller designs known to a person having ordinary skill in the art may be used.

(21) While the present disclosure focuses on trimming out the bias torque caused by the CoG offset, it should be appreciated, as discussed above, that the same arrangement can be applied to a multitude of causes that generate unwanted bias torque.

(22) If there is no bias torque (caused by a non-zero CoG offset or any other source), all rotor thrusts should be about the same when the QR 100 is in steady state operations (e.g., steadily hovering or low-speed uniform motion). In such a case, the QR has the best performance.

(23) Once an eccentric payload 148 is loaded on the QR 100, the CoG 171 of the QR 100 (together with the payload 148) is fixed but shifts from the geometric center of the rotors 152.sub.i of the QR 100 to a second CoG causing a CoG offset. When this occurs, the geometry of the QR 100, defined by the locations of the four rotors 152.sub.i, changes with the rotation of the rotatable arms 142.sub.i. Therefore, the arrangement disclosed herein is designed to adjust the geometric center of the rotors 152.sub.i of the QR 100 (via rotatable arms 142.sub.i) such that the geometric center of the rotors 152.sub.i of the QR 100 can be moved to where a second CoG is projected on the plane formed by the four rotors 152.sub.i after the QR 100 has acquired a payload 48. If there is a bias torque (caused by whatever source), rotor thrusts vary in order to fight the bias torque, which reduces the performance of the QR 100. In this case, the geometric center of the rotors 152.sub.i of the QR 100 should be adjusted based on the teachings of the present disclosure by rotating the rotatable arms 42 (i.e., by changing the positions of the rotors 52 and associated thrusts). Rotation of the rotatable arms 142 will have an impact on rotor thrusts, namely, by rotating the rotatable arms 142 all rotor thrusts tend to converge to the same thrust output (¼ of the total weight of the quadrotor 100, the payload 148, and/or other external forces exerted on the quadrotor 100).

(24) The equations and vectors detailed below are set forth in a Cartesian coordinate system. However, cylindrical and spherical coordinate systems may be used to substitute the existing vector notation where appropriate. Such substitution can be made by methods known to those having ordinary skill in the art.

(25) In order for the Flight Trim Controller 61 to control each of the rotatable arms 42 to a selective position, it needs 1) a dynamics model of a QR 100 with the bias torque caused by the CoG offset taken into account, and 2) a flight trim system using the dynamics model to trim out the bias torque caused by the CoG offset. To establish the model, the angular positions of ζ.sub.1, ζ.sub.2, ζ.sub.3, and ζ.sub.4, with each rotatable arm 142.sub.i terminating with a rotor 152.sub.i having a thrust force F.sub.1, F.sub.2, F.sub.3, and F.sub.4, respectively, are used. While the QR 100 shown in FIG. 2 is of a QRA variety (i.e., rotatable arms with selective angular position), in a QFA variety, the angle of each arm of the QR is fixed as ζ.sub.ƒ1, ζ.sub.ƒ2, ζ.sub.ƒ3, and ζ.sub.ƒ4, respectively. Therefore, while ζ.sub.i (i=1, 2, 3, 4) are variables, ζ.sub.ƒi (i=1, 2, 3, 4) are constants ([ζ.sub.ƒ1 ζ.sub.ƒ2 ζ.sub.ƒ3 ζ.sub.ƒ4].sup.T=[45° 135°-135°-45°].sup.T). In FIG. 2 or 3, each of the rotatable arms 142.sub.i has a length “a” which is defined as the distance between a vertical axis running through the center of each rotor 152.sub.i to the z-axis, where a can be different for each rotatable arm 142.sub.i. The CoG 171 is defined by l.sub.g=[l.sub.x l.sub.y l.sub.z].sup.T. The force of gravity operating on the QR 100 is F.sub.g=[F.sub.x F.sub.y F.sub.z].sup.T which has a magnitude mg where m is the total mass of the QR 100 plus payload 48 and g is the gravitational constant.

(26) The bias torque τ.sub.g is generally expressed as

(27) $\begin{array}{cc}{\tau }_g=l_g\times F_g\mathrm{or}\left[\begin{array}{c}{\tau }_{\mathrm{gx}}\\ {\tau }_{\mathrm{gy}}\\ {\tau }_{\mathrm{gz}}\end{array}\right]=\left[\begin{array}{c}{l}_y{F}_z-l_z{F}_y\\ l_z{F}_x-l_x{F}_z\\ l_x{F}_y-l_y{F}_x\end{array}\right]& \left(1\right)\end{array}$
The rotor thrust F.sub.i (i=1, 2, 3, 4), the magnitude of the total thrust F.sub.r and the total torque generated by all rotor thrust τ.sub.r, denoted by τ.sub.r=[τ.sub.rx τ.sub.ry τ.sub.rz].sup.T, have the following relationship:

(28) $\begin{array}{cc}\left[\begin{array}{c}{F}_1\\ F_2\\ F_3\\ F_4\end{array}\right]=R\left[\begin{array}{c}{\tau }_{\mathrm{rx}}\\ {\tau }_{\mathrm{ry}}\\ {\tau }_{\mathrm{rz}}\\ F_r\end{array}\right]& \left(2\right)\end{array}$
where R is the rotation matrix 267 defined as provided below:

(29) $\begin{array}{cc}R={\left[\begin{array}{cccc}a\sin {\zeta }_1& a\sin {\zeta }_2& a\sin {\zeta }_3& a\sin {\zeta }_4\\ -a\cos {\zeta }_1& -a\cos {\zeta }_2& -a\cos {\zeta }_3& -a\cos {\zeta }_4\\ -\frac{{\tau }_{d1}}{F_1}& -\frac{{\tau }_{d2}}{F_2}& -\frac{{\tau }_{d3}}{F_3}& -\frac{{\tau }_{d4}}{F_4}\\ 1& 1& 1& 1\end{array}\right]}^{-1}& \left(3\right)\end{array}$
where τ.sub.di (i=1, 2, 3, 4) represents the aerodynamic drag of the ith rotor 152.sub.i and τ.sub.di is proportional to the thrust (F.sub.i) of the ith rotor 152.sub.i. A constant of proportionality between τ.sub.di and F.sub.i is about 0.02.

(30) The angular positions of each rotatable arm 142.sub.i (i.e., ζ.sub.1, ζ.sub.2, ζ.sub.3, and ζ.sub.4) can be solved by satisfying the two following equations.

(31) $\hspace{1em}\{\begin{array}{cc}{\tau }_{\mathrm{gx}}+a\underset{i=1}{\overset{4}{.Math.}}\sin {\zeta }_i{F}_i=0& \left(4\right)\\ {\tau }_{\mathrm{gy}}+a\underset{i=1}{\overset{4}{.Math.}}\cos {\zeta }_i{F}_i=0& \left(5\right)\end{array}$
Here there are four unknowns (i.e., ζ.sub.1, ζ.sub.2, ζ.sub.3, and ζ.sub.4) and two equations (4) and (5). Therefore, one can have infinite sets of solutions. In order to obtain a unique solution, an objective function is introduced

(32) $\begin{array}{cc}f\left({\zeta }_n\right)=\underset{i=1}{\overset{4}{.Math.}}.Math.{\zeta }_i-{\zeta }_{\mathrm{fi}}.Math.& \left(6\right)\end{array}$
to further constrain the solution, where ζ.sub.ƒi (i=1, 2, 3, 4) represents the arm-associated angle of the ith arm of a QFA at [ζ.sub.ƒ1 ζ.sub.ƒ2 ζ.sub.ƒ3 ζ.sub.ƒ4].sup.T=[45° 135°-135°-45°].sup.T.

(33) With the objective function (6) achieving its minimum, the total change of arm-associated angles of the proposed QR 100 from the corresponding QFA's constant arm-associated angles is the least. It should be appreciated that the above objective function (6) is just an example. One can come up with other constraints to obtain a unique solution of the arm-associated angles.

(34) The schematic depicts the CoG 171 shifted from the origin of the coordinate axis to a location defined by coordinates l.sub.g=[l.sub.x l.sub.y l.sub.z].sup.T which is defined above. The plane in which all four of the rotatable arms 142.sub.i rotate within is the x-y plane of F.sub.B. The vertical axis with which the rotatable arms 142.sub.i rotate about is normal to the x-y plane of F.sub.B. Rotor thrusts F.sub.1, F.sub.2, F.sub.3, and F.sub.4 are individually generated by the corresponding rotors 152.sub.i and are shown normal to the x-y plane of F.sub.B. Rotor thrusts need not be equal.

(35) With reference to FIG. 4, a flowchart 200 is provided to explain the actions of the Flight Trim Controller 261 (same as the Flight Trim Controller 61 of FIG. 1) as well as the Flight Controller 260 (same as the Flight Controller 60 of FIG. 1), reference to both of which were made in FIG. 1. A position controller 269 and an attitude controller 268 of a QR 100 which has rotatable arms 142.sub.i, according to FIG. 3, is also shown in FIG. 4.

(36) Initially the Flight Controller 260 receives flight data from a subsystem based on manual input or external data which can include attitude and speed requirements as referenced in FIG. 1. The flight data is then processed and converted into a target total thrust and target attitude which is controlled by the position controller 269. The target total thrust and target attitude are sent to the attitude controller 268, which then determines target torques. The target torques and target total thrust are mapped to the target individual rotor thrusts using the rotation matrix 267.

(37) The stability is determined if the absolute value of the angular velocity of the QR 100 is less than a predetermined threshold. Depending on applications, a stable attitude may be defined based on different criteria (e.g., when the attitude error, angular velocity, and angular acceleration of the QR 100 are less than specific thresholds).

(38) When the attitude of the QR 100 is unstable (i.e., the angular velocity of the QR 100 is greater than a predetermined threshold), the Flight Trim Controller 261 keeps the rotatable arms 142.sub.i at their current position. The reason for this approach is that it will become exceedingly complex to adjust the position of the rotatable arms 142.sub.i when the QR 100 is unstable. If the Flight Trim Controller 261 determines the QR 100 is stable (i.e., the angular velocity of the QR 100 is equal or less than a predetermined threshold), then it calculates the bias torque as provided above. Once the bias torque has been calculated, the Flight Trim Controller 261 calculates the target arm-associated angles based on the calculated bias torque, as provided above. Once the target positions of the rotatable arms 142.sub.i have been calculated, the Flight Trim Controller 261 activates the associated actuators (e.g., servo 1 to servo n in FIG. 1) to rotate all rotatable arms 142.sub.i to new positions defined by the calculated target arm-associated angles. Based on the current arm-associated angles of the QR 100, the rotation matrix 267 is updated.

(39) The current position and attitude data of the QR 100 are then fed back to the respective position controller 269 and attitude controller 268 to transform the target total thrust and the target torques about three axes x, y, and z to four target individual rotor thrusts using the newly updated rotation matrix 267 to control the position and attitude of the QR 100 in three-dimensional space.

(40) Those having ordinary skill in the art will recognize that numerous modifications can be made to the specific implementations described above. The implementations should not be limited to the particular limitations described. Other implementations may be possible.