CN111487996B - Multi-unmanned aerial vehicle cooperative control system based on ADRC control and method thereof - Google Patents
Multi-unmanned aerial vehicle cooperative control system based on ADRC control and method thereof Download PDFInfo
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Abstract
The invention discloses a multi-unmanned aerial vehicle cooperative control system based on ADRC control and a method thereof. The system comprises a master unmanned aerial vehicle, slave unmanned aerial vehicles, a parameter server, a master unmanned aerial vehicle joint state issuing module M0, a master unmanned aerial vehicle deviation operation module M1, a master unmanned aerial vehicle position controller response module M2, an ADRC controller-based slave unmanned aerial vehicle following control compensation module M3 and a slave unmanned aerial vehicle position controller response module M4. The invention comprehensively utilizes theories and methods such as unmanned aerial vehicle model building, double-unmanned aerial vehicle cooperative control simulation, ADRC control and the like under a Gazebo simulation platform, and solves the key technical problems of stable, fast response speed and high-efficiency real-time follow-up control of slave unmanned aerial vehicles to a master unmanned aerial vehicle in a multi-unmanned aerial vehicle cooperative control system under different working environments.
Description
Technical Field
The invention relates to the field of unmanned aerial vehicle control, in particular to a multi-unmanned aerial vehicle cooperative control system based on ADRC control and a method thereof.
Background
With the development of scientific technology and the maturity of scientific theory, more and more high-tech products come into thousands of households gradually. Like unmanned aerial vehicle, the unmanned aerial vehicle of models such as domestic type, amusement type and specialty type has been developed by many large-scale enterprises now and has been supplied ordinary consumer to use, and these models easy operation, flight stability safety, some are from taking the camera in addition. The unmanned aerial vehicle can be developed rapidly because of small volume, rapid movement and flexible control, and can be applied to media shooting, medical equipment transfer, wounded rescue, dangerous airspace detection, military reconnaissance and anti-reconnaissance, visual arts of multi-unmanned aerial vehicle cooperative formation and the like.
At present, with the vigorous development of unmanned aerial vehicle technology, the emerging industry of unmanned aerial vehicle cargo transportation has gone into thousands of households like the spring shoots after rain. However, the relatively mature unmanned aerial vehicle is light and light, has a small load, can carry no more than 10 kg of articles (the physical distribution development overview of unmanned aerial vehicle-Lilian), and cannot meet the transportation requirements of people. If the size of the unmanned aerial vehicle is simply enlarged, although the load capacity can be improved, the flexibility of the flight is greatly reduced. The size of the unmanned aerial vehicle with the load of more than 50 kilograms can reach more than 1m (application analysis of the multi-rotor unmanned aerial vehicle in tactical logistics — Tiggui). Therefore, people begin to aim at the field of cooperative control of multiple unmanned aerial vehicles, the working efficiency and fault tolerance of the whole unit can be improved through cooperative work of the multiple unmanned aerial vehicles, and many mechanisms strive to realize a cooperative control system of the multiple unmanned aerial vehicles.
In order to fully develop drones, people are beginning to continuously research and improve the traditional drone control algorithm. Through many unmanned aerial vehicle cooperative control based on auto-disturbance rejection controller ADRC, not only made the improvement to traditional linear PID controller in control algorithm, promoted the development that the unmanned aerial vehicle nonlinear control theory was used, further try to control many unmanned aerial vehicles simultaneously, let unmanned aerial vehicle's flight performance more superior to adopt the mode of many unmanned aerial vehicle cooperative work, can improve unmanned aerial vehicle's work efficiency, let the working method nimble more changeable.
Disclosure of Invention
In view of the above, the invention provides a multi-unmanned aerial vehicle cooperative control system based on ADRC control and a method thereof, which comprehensively use theories and methods of unmanned aerial vehicle model building, double-unmanned aerial vehicle cooperative control simulation, ADRC control and the like under a Gazebo simulation platform, and solve the key technical problems of efficient cooperative work and stable flight when a plurality of unmanned aerial vehicles cooperatively work.
The purpose of the invention is realized by at least one of the following technical solutions.
A multi-unmanned aerial vehicle cooperative control system based on ADRC control comprises a master unmanned aerial vehicle, slave unmanned aerial vehicles, a parameter server, a master unmanned aerial vehicle joint state issuing module M0, a master unmanned aerial vehicle deviation operation module M1, a master unmanned aerial vehicle position controller response module M2, a slave unmanned aerial vehicle following control compensation module M3 based on ADRC controllers, and a slave unmanned aerial vehicle position controller response module M4; wherein, a load is connected between the master unmanned aerial vehicle and the slave unmanned aerial vehicle;
the main unmanned aerial vehicle joint state issuing module M0 is used for issuing the state of each joint of the expected position of the main unmanned aerial vehicle to the parameter server and transmitting the state to the main unmanned aerial vehicle deviation operation module M1;
the main unmanned aerial vehicle deviation operation module M1 is used for receiving the parameters of the M0 module, carrying out corresponding operation to obtain deviation, and outputting the control action of the controller, namely the rotor speed, to the main unmanned aerial vehicle position controller response module M2 according to the deviation;
the master unmanned aerial vehicle position controller response module M2 is configured to obtain a corresponding rotor speed, so that each rotor of the master unmanned aerial vehicle makes a corresponding response to move to a desired position;
the slave unmanned aerial vehicle following control compensation module M3 based on the ADRC controller obtains the passive force borne by the slave unmanned aerial vehicle and transmitted by the load according to the rotating speed of each rotor of the master unmanned aerial vehicle, and outputs expected acceleration a 'describing the slave unmanned aerial vehicle'des；
The position controller of the slave drone outputs the rotor speed ω 'of the slave drone in response to the module M4 through the desired acceleration in the input M3'rotorAnd the control of the slave unmanned aerial vehicle is realized.
Further, the master unmanned aerial vehicle joint state issuing module M0 transmits the real-time state of each joint including coordinates, linear velocity and angular velocity to the parameter server.
Further, in the master unmanned aerial vehicle deviation operation module M1: the node of the controller of the main unmanned aerial vehicle calculates according to the real-time state parameters of the main unmanned aerial vehicle to obtain deviation and outputs the control action of the position controller, namely the corresponding rotating speed of each motor; the real-time state parameters of the main unmanned aerial vehicle comprise the coordinates, linear speed and angular speed of the main unmanned aerial vehicle.
And the main unmanned aerial vehicle position controller response module M2 operates the motors according to the rotating speeds corresponding to the motors output by the main unmanned aerial vehicle deviation operation module M1.
A multi-unmanned aerial vehicle cooperative control method based on ADRC control comprises the following steps:
s1, a main unmanned aerial vehicle joint state publishing module M0 publishes the state of each joint of the expected position of the main unmanned aerial vehicle to a parameter server, and transmits the state to a main unmanned aerial vehicle deviation operation module M1;
s2, after receiving the parameters of the master unmanned aerial vehicle joint state release module M0, the master unmanned aerial vehicle deviation operation module M1 performs corresponding operation to obtain deviation, and outputs the control action of the controller, namely the rotor speed, to the master unmanned aerial vehicle position controller response module M2 according to the deviation;
s3, the main unmanned aerial vehicle position controller response module M2 obtains corresponding rotor rotating speed, so that each rotor of the main unmanned aerial vehicle makes a corresponding response to move to a desired position, at the moment, the real-time state of each joint changes, the real-time state parameters of the main unmanned aerial vehicle are transmitted to the main unmanned aerial vehicle deviation operation module M1, the main unmanned aerial vehicle deviation operation module M1 obtains the control function of a deviation output controller through operation and transmits the control function to the main unmanned aerial vehicle position controller response module M2, and a complete negative feedback control loop is formed;
s4, the slave unmanned aerial vehicle following control compensation module M3 based on the ADRC calculates the passive force borne by the slave unmanned aerial vehicle and transferred by the load through an ADRC algorithm, and outputs a ' describing the expected acceleration a ' of the slave unmanned aerial vehicle 'des；
S5, the slave drone 'S position controller response module M4 outputs the slave drone' S rotor speed ω 'by inputting the desired acceleration in the ADRC controller based slave drone following control compensation module M3'rotorAnd the control of the slave unmanned aerial vehicle is realized.
Further, the method for cooperative control of multiple drones based on ADRC control according to claim 5, wherein step S2 includes the following steps:
s2.1, receiving real-time state parameters of the main unmanned aerial vehicle, including coordinates, linear speed and angular speed, issued by a joint state issuing module M0 of the main unmanned aerial vehicle by a deviation operation module M1 of the main unmanned aerial vehicle;
s2.2, after the master unmanned aerial vehicle controller node obtains the parameters in the step S2.1, deviation calculation is carried out to obtain deviation based on the deviation, and the control action of the position controller, namely the corresponding rotating speed of each motor, is output;
s2.3, each rotor of the main unmanned aerial vehicle receives the control action to make a corresponding response, and the real-time state of each joint changes.
Further, in step S2.2, the control function of obtaining the deviation based on the deviation calculation and outputting the deviation to the position controller, that is, the corresponding rotation speed of each motor is specifically as follows:
three dimensional column vector gn_attFor normalized attitude gain, a three-dimensional row vector gattFor attitude gain in the parameter file, the rotational inertia of the unmanned aerial vehicle is imn(m, n ═ x, y, z), then
Wherein ixx＝∫∫∫V(y2+z2)ρdV；iyy＝∫∫∫V(x2+z2)ρdV；izz＝∫∫∫V(x2+y2)ρdV；ixy＝∫∫∫V(xy)ρdV；ixz＝∫∫∫V(xz)ρdV；iyz＝∫∫∫V(yz) ρ dV; (x, y, z are three directions of coordinate axes, ixx、iyy、izzThe moments of inertia for the x, y, and z axes, ixy、ixz、iyzIs product of inertia)
Three dimensional column vector gn_angFor the normalized angular velocity gain, the angular velocity gain in the parameter file of the parameter server is gangThen there is
Let the allocation matrix be a and,
the inertia matrix I is
The angular acceleration a of the rotor speedrv_angIs provided with
arv_ang＝AT*(A*AT)-1*I； (5)
Let the three-dimensional column vector position error be PerrSpecifying the track position as PctThe actual position is PactThen there is
Perr＝Pact-Pct； (6)
Let the three-dimensional column vector velocity error be verrSpecifying a trajectory velocity vctThe actual speed is vactThen there is
verr＝vact-vct； (7)
Let the unmanned aerial vehicle mass be m, the position gain in the parameter file of the parameter server be gposThe velocity gain is gvelG is gravity acceleration, and a is trajectory accelerationctThen the desired acceleration adesIs composed of
If the elevation angle of the unmanned aerial vehicle is yaw, the specific parameters are as follows:
b3des＝-ades*norm(ades)； (10)
b2des＝b3des×b1des； (11)
wherein, b1des、b3des、b2desFor tracking yaw angleReference trajectory signal, yaw, represents the yaw angle of the drone, norm (a)des) Denotes ades2-norm of (d); the three-dimensional expected positioning matrix R can be obtaineddesIs composed of
Rdes＝[b2des×b3des b2des b3des]； (12)
Let the three-dimensional angle error matrix be AngerrThe positioning matrix is a matrix of R,
the three-dimensional column vector angle error angerrIs (Ang)err(a, b) represents AngerrMatrix a row b column element)
Let the desired angular rate be ratedes_angThe actual angular rate is rateact_angThen angular rate error rateerr_angIs composed of
Angular acceleration of arateThen there is
arate＝-angerr·gn_att-rateerr_ang·gn_ang+rateact_ang×rateact_ang；
(17)
If the thrust in the z-axis direction is t and the mass of the main unmanned aerial vehicle is m, the
t＝-[0 0 m]*ades； (18)
Then the four-dimensional column vector angular acceleration a of the thrust t is introducedrate_tIs composed of
The final controller obtains the rotor speed omegarotorIs composed of
ωrotor＝arv_ang*arate_t； (20)
arv_angRepresenting the rotor speed angular acceleration.
Further, step S4 specifically includes the following steps:
s4.1, according to the speed v of the slave unmanned aerial vehicle in the directions of the x axis and the y axisxAnd vyCalculating the magnitude f of the driven force in the directions of the x and y axesxAnd fyThe specific calculation is as follows:
wherein v isx0And vy0Are each t0Velocity v of time-dependent unmanned aerial vehicle in x and y axis directionsx1And vy1Are each t1The speed of the moment in the x and y axis directions, m is the mass of the unmanned aerial vehicle, fxAnd fyAt t for the slave unmanned aerial vehicle respectively0To t1The magnitude of the passive force received in the time in the directions of the x axis and the y axis;
s4.2, converting the f obtained in the step S4.1xAnd fyThe acceleration control quantity u in the x-axis direction and the y-axis direction is obtained by transmitting the acceleration control quantity u into an ADRC controllerxAnd uy；
S4.3, according to the acceleration control quantity u in the step S4.2xAnd uyUpdating a 'desired acceleration of slave unmanned'des。
Further, in step S4.2, the ADRC controller includes a transition process, a nonlinear combination and an Extended State Observer (ESO), b0Is a rough estimate of the control gain of the controlled object motor and the slave drone, i.e. the compensation factors for the ESO and disturbance compensation;
the transition process carries out prediction processing on the real-time expected acceleration, and the controlled variable tracks the given input to obtain v1、v2Wherein v is0Is a control purpose, v1Is v0Transition process of v2Is v0Respectively for the difference calculation of the subsequent non-linear combination; in order to track a given input at the fastest speed, a fastest control synthesis function fhan (x) is introduced1,x2R, h), the expression of which is shown in formula (23):
thus, the entire transition process is represented as follows:
wherein sign () is a sign function; r is0The parameter is one of the parameters of the controller, and is specifically adjusted according to the time required by the transition process of the system; h is the system sampling step size, which mainly affects beta in ESO01、β02、β03Parameter, beta01、β02、β03The specific parameter value of (2) is determined by the sampling process;
the ESO compensates for system-controlled disturbances, i.e. the slave drone rotor speed input u (via b)0Amplification) and following the output of the control process z is obtained by the ESO algorithm1And z2And a total disturbance z of the system equivalent to the input side3Wherein z is1And z2For determining the tracking error and its derivative, z3For directly compensating the disturbance, thereby accurately estimating the real-time value of the system operation and eliminating the influence of the disturbance(ii) a The specific equation for ESO is as follows:
wherein beta is01、β02And beta03The specific value of the parameter of the controller is determined by the sampling step length h; the fal () function is:
δ () is a unit pulse function and sign () is a sign function; b0Is a compensation factor;
obtaining u by nonlinear combination0I.e. v obtained by the transition element is separately measured1(t) and v2(t) and z by extended state observer1And z2Are subtracted to obtain e1And e2Then e is added1And e2Nonlinear combination is carried out to obtain a control quantity u0The nonlinear combination formula is as follows:
wherein r is a controlled variable gain, and when the controlled variable gain exceeds 100, the influence on the system is greatly reduced, so that the value is generally not more than 100; h is1For the accuracy factor, increase h1The corresponding overshoot of the output can be reduced, and the adjusting time is increased, and the reciprocal of the adjusting time is similar to the proportional gain; c is a damping factor, increasing c reduces the settling time of the system, i.e., the response is quicker, similar to differential gain; disturbance compensation of final system control, namely, the rotor speed input of the slave unmanned aerial vehicle is generally in a fixed mode, and the expression is as follows:
as can be seen from equation (27), the final controlled variable includes the feedback controlled variable u0And a compensation control quantity z3Compared with the traditional linear PID, the active disturbance rejection controller developed on the basis of the nonlinear PID regulator can achieve better control effect and improve the performance of a control system.
Further, in step S4.3, the desired acceleration a 'of the slave drone is updated'desThe following were used:
in formulae (28) and (29), ux、uyControl quantities a 'of slave unmanned aerial vehicles in x-axis and y-axis directions'desIs the slave drone expected acceleration; u. of0Is controlled by the fastest control synthesis function fhan (x)1,x2R, h) control quantity of output, b0Is the ADRC controller compensation factor, z3Is a state variable of the extended state observer; the updated acceleration a'desAs the expected acceleration of the slave unmanned aerial vehicle following control, and the updated value a 'at that time'desAs a control target of the rotor speed of the slave unmanned aerial vehicle, the real-time stress of the slave unmanned aerial vehicle is adjusted through the motor.
Further, in step S5, the desired acceleration in M3 is input, and the rotor rotation speed ω 'of the slave drone is output'rotorThe method comprises the following specific steps:
let the elevation angle of the slave unmanned aerial vehicle be yaw', the specific parameters are as follows:
b3′des＝-a′des*norm(a′des)； (31)
b2′des＝b3′des×b1′des； (32)
wherein, b1′des、b3′des、b2′desTracking a reference trajectory signal for yaw angle, norm (a'des) Is a'des2-norm of (d); three-dimensional expected positioning matrix R 'can be obtained'desIs composed of
R′des＝[b2′des×b3′des b2′des b3′des]； (33)
Let three-dimensional angle error matrix be Ang'errThe positioning matrix is R', has
Then three-dimensional column vector angular error ang'errIs (Ang'err(a, b) represents Ang'errMatrix a row b column element)
Let desired angular velocity be rat'des_angActual angular rate is rat'act_angThen angular rate error rate'err_angIs composed of
Let the angular acceleration be a'rateThen there is
a′rate＝-ang′err·g′n_att-rate′err_ang·g′n_ang+rate′act_ang×rate′act_ang；
(38)
If the thrust in the z-axis direction is t 'and the mass of the slave unmanned aerial vehicle is m', then
t′＝-[0 0 m′]*a′des； (39)
Then four-dimensional column vector angular acceleration a 'of thrust t is introduced'rate_tIs composed of
The final controller obtains the rotor speed omega of the slave unmanned aerial vehicle'rotorIs omega'rotor＝a′rv_ang*a′rate_t； (41)
a′rv_angRepresenting the rotor speed angular acceleration.
Compared with the prior art, the invention has the advantages that:
the invention comprehensively utilizes theories and methods of unmanned aerial vehicle model building, double-unmanned aerial vehicle cooperative control simulation, ADRC control, Simulink simulation and the like under a Gazebo simulation platform, adopts a cooperative working mode of multiple unmanned aerial vehicles, improves the working efficiency of the unmanned aerial vehicles, and ensures that the working mode is more flexible and changeable. Meanwhile, load bearing can be shared, the load capacity and the lowest performance requirement of a battery of a single unmanned aerial vehicle are reduced, the carrying process is more gentle, the path is more flexible, and when cooperative matching is not needed, the unmanned aerial vehicle units can work in a dispersed and independent mode, and the service performance and the working efficiency of the unmanned aerial vehicle units are improved. Meanwhile, when the cooperative unmanned aerial vehicle breaks down, the cooperative unmanned aerial vehicle can be replaced in time, and the influence on the whole unit is reduced to the minimum; compared with the traditional multi-unmanned-plane cooperative control system, the multi-unmanned-plane cooperative control system has the advantages that the communication between the master unmanned plane and the slave unmanned planes is not realized through a wireless network communication link any more, but the direction and the magnitude of the passive force transmitted by the load on the slave unmanned planes are estimated, the passive force is compensated, so that the external force on the slave unmanned planes is zero, and the slave unmanned planes can realize the real-time following of the master unmanned planes; when controlling the slave unmanned aerial vehicle, realize following control with the position controller that adopts specific parameter and compare, the flight path of main unmanned aerial vehicle is more accurate among the follow control system of utilizing the auto-disturbance rejection controller to realize slave unmanned aerial vehicle, and slave unmanned aerial vehicle can realize following at a higher speed simultaneously, and control effect is better.
Drawings
Fig. 1 is a schematic overall structure diagram of a multi-drone cooperative control system based on ADRC control according to the present invention;
fig. 2 is a schematic structural diagram of a control principle of a master unmanned aerial vehicle in the embodiment of the present invention;
fig. 3 is a schematic structural diagram of a control principle of a slave drone in the embodiment of the present invention.
Detailed Description
The following description will further explain embodiments of the present invention by referring to the figures and examples.
Example (b):
the invention uses the position controller to realize the required fixed value control on the main unmanned aerial vehicle, and achieves the purpose of fixed value control by eliminating the deviation between the expected position coordinates and the measured position coordinates; the utility model provides a many unmanned aerial vehicle cooperative control system with passive power as controlled variable, exchange between master unmanned aerial vehicle and the subordinate unmanned aerial vehicle is through predicting the direction and the size of the passive power that receives by the load that subordinate unmanned aerial vehicle received, compensate to passive power and make the external force that subordinate unmanned aerial vehicle receives be zero, realize subordinate unmanned aerial vehicle and follow master unmanned aerial vehicle in real time, the problem that subordinate unmanned aerial vehicle probably got into out of control state when traditional many unmanned aerial vehicle cooperative control appeared signal interruption is solved, the dependence of principal and subordinate unmanned aerial vehicle exchange to communication speed and signal quality has been eliminated. Meanwhile, the ADRC controller is adopted to realize the following control of the slave unmanned aerial vehicle, so that the effects that the flight path of the master unmanned aerial vehicle is more accurate and the slave unmanned aerial vehicle follows at a higher speed are realized. The ADRC controller, namely Active Disturbance Rejection Control, has strong robustness and adaptability, can perform pre-estimation compensation on unknown interference to overcome the Disturbance, inherits the stability, rapidity and accuracy of the traditional linear PID controller in performance, and introduces nonlinear combination to adapt to more complex Control objects.
An ADRC control-based multi-unmanned aerial vehicle cooperative control system is shown in figure 1 and comprises a master unmanned aerial vehicle, slave unmanned aerial vehicles, a parameter server, a master unmanned aerial vehicle joint state issuing module M0, a master unmanned aerial vehicle deviation operation module M1, a master unmanned aerial vehicle position controller response module M2, an ADRC controller-based slave unmanned aerial vehicle following control compensation module M3 and a slave unmanned aerial vehicle position controller response module M4; wherein, a load is connected between the master unmanned aerial vehicle and the slave unmanned aerial vehicle;
the main unmanned aerial vehicle joint state issuing module M0 is used for issuing the state of each joint of the expected position of the main unmanned aerial vehicle to the parameter server and transmitting the state to the main unmanned aerial vehicle deviation operation module M1;
the main unmanned aerial vehicle deviation operation module M1 is used for receiving the parameters of the M0 module, carrying out corresponding operation to obtain deviation, and outputting the control action of the controller, namely the rotor speed, to the main unmanned aerial vehicle position controller response module M2 according to the deviation;
the master unmanned aerial vehicle position controller response module M2 is configured to obtain a corresponding rotor speed, so that each rotor of the master unmanned aerial vehicle makes a corresponding response to move to a desired position;
the slave unmanned aerial vehicle following control compensation module M3 based on the ADRC controller obtains the passive force which is borne by the slave unmanned aerial vehicle and is transmitted by the load according to the rotating speed of each rotor of the master unmanned aerial vehicle, and outputs the expected acceleration a which describes the slave unmanned aerial vehicledes；
The position controller of the slave drone responds to the module M4 by outputting the rotor speed ω of the slave drone through the desired acceleration in the input M3rotorAnd the control of the slave unmanned aerial vehicle is realized.
The master unmanned aerial vehicle joint state publishing module M0 transmits the real-time state of each joint including coordinates, linear speed and angular speed to the parameter server.
In the master unmanned aerial vehicle deviation operation module M1: the node of the controller of the main unmanned aerial vehicle calculates according to the real-time state parameters of the main unmanned aerial vehicle to obtain deviation and outputs the control action of the position controller, namely the corresponding rotating speed of each motor; the real-time state parameters of the main unmanned aerial vehicle comprise the coordinates, linear speed and angular speed of the main unmanned aerial vehicle.
And the main unmanned aerial vehicle position controller response module M2 operates the motors according to the rotating speeds corresponding to the motors output by the main unmanned aerial vehicle deviation operation module M1.
A method for cooperative control of multiple unmanned aerial vehicles based on ADRC control is disclosed, as shown in FIG. 1, and includes the following steps:
s1, a main unmanned aerial vehicle joint state publishing module M0 publishes the state of each joint of the expected position of the main unmanned aerial vehicle to a parameter server, and transmits the state to a main unmanned aerial vehicle deviation operation module M1;
s2, after receiving the parameters of the master unmanned aerial vehicle joint state release module M0, the master unmanned aerial vehicle deviation operation module M1 performs corresponding operation to obtain deviation, and outputs the control action of the controller, namely the rotor speed, to the master unmanned aerial vehicle position controller response module M2 according to the deviation; as shown in fig. 2, the method comprises the following steps:
s2.1, receiving real-time state parameters of the main unmanned aerial vehicle, including coordinates, linear speed and angular speed, issued by a joint state issuing module M0 of the main unmanned aerial vehicle by a deviation operation module M1 of the main unmanned aerial vehicle;
s2.2, after the master unmanned aerial vehicle controller node obtains the parameters in the step S2.1, deviation calculation is carried out to obtain deviation based on the deviation, and the control action of the position controller, namely the corresponding rotating speed of each motor, is output;
the control function of obtaining the deviation based on the deviation calculation and outputting the deviation to the position controller, namely the corresponding rotating speed of each motor, is as follows:
three dimensional column vector gn_attFor normalized attitude gain, a three-dimensional row vector gattFor attitude gain in the parameter file, the rotational inertia of the unmanned aerial vehicle is imn(m, n ═ x, y, z), then
Wherein ixx＝∫∫∫V(y2+z2)ρdV；iyy＝∫∫∫V(x2+z2)ρdV；izz＝∫∫∫V(x2+y2)ρdV；ixy＝∫∫∫V(xy)ρdV；ixz＝∫∫∫V(xz)ρdV；iyz＝∫∫∫V(yz) ρ dV; (x, y, z are three directions of coordinate axes, ixx、iyy、izzThe moments of inertia for the x, y, and z axes, ixy、ixz、iyzIs product of inertia)
Three dimensional column vector gn_angFor the normalized angular velocity gain, the angular velocity gain in the parameter file of the parameter server is gangThen there is
Let the allocation matrix be a and,
the inertia matrix I is
The angular acceleration a of the rotor speedrv_angIs provided with
arv_ang＝AT*(A*AT)-1*I； (5)
Let the three-dimensional column vector position error be PerrSpecifying the track position as PctThe actual position is PactThen there is
Perr＝Pact-Pct； (6)
Let the three-dimensional column vector velocity error be verrSpecifying a trajectory velocity vctThe actual speed is vactThen there is
verr＝vact-vct； (7)
Let the unmanned aerial vehicle mass be m, the position gain in the parameter file of the parameter server be gposThe velocity gain is gvelG, gravity acceleration, and specifying railTrace acceleration of actThen the desired acceleration adesIs composed of
If the elevation angle of the unmanned aerial vehicle is yaw, the specific parameters are as follows:
b3des＝-ades*norm(ades)； (10)
b2des＝b3des×b1des； (11)
wherein, b1des、b3des、b2desTracking a reference trajectory signal for yaw, yaw representing the yaw of the drone, norm (a)des) Denotes ades2-norm of (d); the three-dimensional expected positioning matrix R can be obtaineddesIs composed of
Rdes＝[b2des×b3des b2des b3des]； (12)
Let the three-dimensional angle error matrix be AngerrThe positioning matrix is a matrix of R,
the three-dimensional column vector angle error angerrIs (Ang)err(a, b) represents AngerrMatrix a row b column element)
Let the desired angular rate be ratedes_angThe actual angular rate is rateact_angThen angular rate error rateerr_angIs composed of
Angular acceleration of arateThen there is
arate＝-angerr·gn_att-rateerr_ang·gn_ang+rateact_ang×rateact_ang；
(17)
Let the thrust in the z-axis direction be t, then
t＝-[0 0 m]*ades； (18)
Then the four-dimensional column vector angular acceleration a of the thrust t is introducedrate_tIs composed of
The final controller obtains the rotor speed omegarotorIs composed of
ωrotor＝arv_ang*arate_t； (20)
arv_angRepresenting the rotor speed angular acceleration.
S2.3, each rotor of the main unmanned aerial vehicle receives the control action to make a corresponding response, and the real-time state of each joint changes.
S3, the main unmanned aerial vehicle position controller response module M2 obtains corresponding rotor rotating speed, so that each rotor of the main unmanned aerial vehicle makes a corresponding response to move to a desired position, at the moment, the real-time state of each joint changes, the real-time state parameters of the main unmanned aerial vehicle are transmitted to the main unmanned aerial vehicle deviation operation module M1, the main unmanned aerial vehicle deviation operation module M1 obtains the control function of a deviation output controller through operation and transmits the control function to the main unmanned aerial vehicle position controller response module M2, and a complete negative feedback control loop is formed;
s4, the slave unmanned aerial vehicle following control compensation module M3 based on the ADRC calculates the passive force borne by the slave unmanned aerial vehicle and transferred by the load through an ADRC algorithm, and outputs a ' describing the expected acceleration a ' of the slave unmanned aerial vehicle 'des(ii) a As shown in fig. 3, the method specifically includes the following steps:
s4.1, according to the speed v of the slave unmanned aerial vehicle in the directions of the x axis and the y axisxAnd vyCalculating the magnitude f of the driven force in the directions of the x and y axesxAnd fyThe specific calculation is as follows:
wherein v isx0And vy0Are each t0Velocity v of time-dependent unmanned aerial vehicle in x and y axis directionsx1And vy1Are each t1The speed of the moment in the x and y axis directions, m is the mass of the unmanned aerial vehicle, fxAnd fyAt t for the slave unmanned aerial vehicle respectively0To t1The magnitude of the passive force received in the time in the directions of the x axis and the y axis;
s4.2, converting the f obtained in the step S4.1xAnd fyThe acceleration control quantity u in the x-axis direction and the y-axis direction is obtained by transmitting the acceleration control quantity u into an ADRC controllerxAnd uy；
In step S4.2, the ADRC controller includes a transition process, a nonlinear combination and an Extended State Observer (ESO), b0Is a rough estimate of the control gain of the controlled object motor and the slave drone, i.e. the compensation factors for the ESO and disturbance compensation;
the transition process carries out prediction processing on the real-time expected acceleration, and the controlled variable tracks the given input to obtain v1、v2Wherein v is0Is a control purpose, v1Is v0Transition process of v2Is v0Differential of (2)Signals respectively used for the difference operation of the subsequent nonlinear combination; in order to track a given input at the fastest speed, a fastest control synthesis function fhan (x) is introduced1,x2R, h), the expression of which is shown in formula (23):
thus, the entire transition process is represented as follows:
wherein sign () is a sign function; r is0The parameter is one of the parameters of the controller, and is specifically adjusted according to the time required by the transition process of the system; h is the system sampling step size, which mainly affects beta in ESO01、β02、β03Parameter, beta01、β02、β03The specific parameter value of (2) is determined by the sampling process; in this example,. beta.01、β02、β03Taking 0.01 as parameters;
the ESO compensates for system-controlled disturbances, i.e. the slave drone rotor speed input u (via b)0Amplification) and following the output of the control process z is obtained by the ESO algorithm1And z2And a total disturbance z of the system equivalent to the input side3Wherein z is1And z2For determining the tracking error and its derivative, z3The method is used for directly compensating the disturbance, thereby accurately estimating a real-time value of the system operation and eliminating the influence of the disturbance; the specific equation for ESO is as follows:
wherein beta is01、β02And beta03In this embodiment, 100, 300, and 1000 are taken as parameters of the controller; the fal () function is:
δ () is a unit pulse function and sign () is a sign function; b0To compensate for the factor, take 0.25, increase b0The overshoot of the output response is reduced, but the response speed is also reduced, similar to the integral gain;
obtaining u by nonlinear combination0I.e. v obtained by the transition element is separately measured1(t) and v2(t) and z by extended state observer1And z2Are subtracted to obtain e1And e2Then e is added1And e2Nonlinear combination is carried out to obtain a control quantity u0The nonlinear combination formula is as follows:
wherein r is a controlled variable gain, and when the controlled variable gain exceeds 100, the influence on the system is greatly reduced, so that the value is generally not more than 100; h is1For the accuracy factor, increase h1The corresponding overshoot of the output can be reduced, and the adjusting time is increased, and the reciprocal of the adjusting time is similar to the proportional gain; c is a damping factor, increasing c reduces the settling time of the system, i.e., the response is quicker, similar to differential gain; disturbance compensation of final system control, namely, the rotor speed input of the slave unmanned aerial vehicle is generally in a fixed mode, and the expression is as follows:
as can be seen from equation (27), the final controlled variable includes the feedback controlled variable u0And a compensation control quantity z3Compared with the traditional linear PID, the active disturbance rejection controller developed on the basis of the nonlinear PID regulator can achieve better control effect and improve the performance of a control system.
S4.3, according to the acceleration control quantity u in the step S4.2xAnd uyUpdating the desired acceleration ad of the slave unmannedes。
As shown in FIG. 3, the update of the desired acceleration a 'of the slave drone'desThe following were used:
in formulae (28) and (29), ux、uyControl quantities a 'of slave unmanned aerial vehicles in x-axis and y-axis directions'desIs the slave drone expected acceleration; u. of0Is controlled by the fastest control synthesis function fhan (x)1,x2R, h) control quantity of output, b0Is the ADRC controller compensation factor, z3Is a state variable of the extended state observer; the updated acceleration a'desAs the expected acceleration of the slave unmanned aerial vehicle following control, and the updated value a 'at that time'desAs a control target of the rotor speed of the slave unmanned aerial vehicle, the real-time stress of the slave unmanned aerial vehicle is adjusted through the motor.
S5, the slave drone 'S position controller response module M4 outputs the slave drone' S rotor speed ω 'by inputting the desired acceleration in the ADRC controller based slave drone following control compensation module M3'rotorAnd the control of the slave unmanned aerial vehicle is realized.
The expected acceleration of the input M3 outputs the rotor speed omega 'of the slave unmanned aerial vehicle'rotorThe method comprises the following specific steps:
let the elevation angle of the slave unmanned aerial vehicle be yaw', the specific parameters are as follows:
b3′des＝-a′des*norm(a′des)； (31)
b2′des＝b3′des×b1′des； (32)
wherein, b 1'des、b3′des、b2′desTracking a reference trajectory signal for yaw angle, norm (a'des) Is a'des2-norm of (d); three-dimensional expected positioning matrix R 'can be obtained'desIs composed of
R′des＝[b2′des×b3′des b2′des b3′des]； (33)
Let three-dimensional angle error matrix be Ang'errThe positioning matrix is R', has
Then three-dimensional column vector angular error ang'errIs (Ang'err(a, b) represents Ang'errMatrix a row b column element)
Let desired angular velocity be rat'des_angActual angular rate is rat'act_angThen angular rate error rate'err_angIs composed of
Let the angular acceleration be a'rateThen there is
a′rate＝-ang′err·g′n_att-rate′err_ang·g′n_ang+rate′act_ang×rate′act_ang；
(38)
If the thrust in the z-axis direction is t 'and the mass of the slave unmanned aerial vehicle is m', then
t′＝-[0 0 m′]*a′des； (39)
Then four-dimensional column vector angular acceleration a 'of thrust t is introduced'rate_tIs composed of
The final controller obtains the rotor speed omega of the slave unmanned aerial vehicle'rotorIs composed of
ω′rotor＝a′rv_ang*a′rate_t； (41)
a′rv_angRepresenting the rotor speed angular acceleration.
The above description is only a preferred embodiment of the present invention, and the present invention is not limited to the above embodiment, but may be used in various other combinations, and there may be some minor structural changes in the detailed implementation, and if various changes or modifications of the present invention do not depart from the spirit and scope of the present invention and fall within the claims and equivalent technical scope of the present invention, the present invention is also intended to include such changes and modifications.
Claims (7)
1. A multi-unmanned aerial vehicle cooperative control method based on ADRC control is characterized in that a multi-unmanned aerial vehicle cooperative control system comprises a master unmanned aerial vehicle, slave unmanned aerial vehicles, a parameter server, a master unmanned aerial vehicle joint state release module M0, a master unmanned aerial vehicle deviation operation module M1, a master unmanned aerial vehicle position controller response module M2, a slave unmanned aerial vehicle following control compensation module M3 based on an ADRC controller and a slave unmanned aerial vehicle position controller response module M4; wherein, a load is connected between the master unmanned aerial vehicle and the slave unmanned aerial vehicle;
the main unmanned aerial vehicle joint state issuing module M0 is used for issuing the state of each joint of the expected position of the main unmanned aerial vehicle to the parameter server and transmitting the state to the main unmanned aerial vehicle deviation operation module M1;
the main unmanned aerial vehicle deviation operation module M1 is used for receiving the parameters of the M0 module, carrying out corresponding operation to obtain deviation, and outputting the control action of the controller, namely the rotor speed, to the main unmanned aerial vehicle position controller response module M2 according to the deviation;
the master unmanned aerial vehicle position controller response module M2 is configured to obtain a corresponding rotor speed, so that each rotor of the master unmanned aerial vehicle makes a corresponding response to move to a desired position;
the slave unmanned aerial vehicle following control compensation module M3 based on the ADRC controller obtains the passive force borne by the slave unmanned aerial vehicle and transmitted by the load according to the rotating speed of each rotor of the master unmanned aerial vehicle, and outputs expected acceleration a 'describing the slave unmanned aerial vehicle'des；
The position controller of the slave drone outputs the rotor speed ω 'of the slave drone in response to the module M4 through the desired acceleration in the input M3'rotorThe control of the slave unmanned aerial vehicle is realized;
the multi-unmanned aerial vehicle cooperative control method comprises the following steps:
s1, a main unmanned aerial vehicle joint state publishing module M0 publishes the state of each joint of the expected position of the main unmanned aerial vehicle to a parameter server, and transmits the state to a main unmanned aerial vehicle deviation operation module M1;
s2, after receiving the parameters of the master unmanned aerial vehicle joint state release module M0, the master unmanned aerial vehicle deviation operation module M1 performs corresponding operation to obtain deviation, and outputs the control action of the controller, namely the rotor speed, to the master unmanned aerial vehicle position controller response module M2 according to the deviation, and the method comprises the following steps:
s2.1, receiving real-time state parameters of the main unmanned aerial vehicle, including coordinates, linear speed and angular speed, issued by a joint state issuing module M0 of the main unmanned aerial vehicle by a deviation operation module M1 of the main unmanned aerial vehicle;
s2.2, after the master unmanned aerial vehicle controller node obtains the parameters in the step S2.1, deviation calculation is carried out to obtain deviation based on the deviation, and the control action of the position controller, namely the corresponding rotating speed of each motor, is output;
the control function of obtaining the deviation based on the deviation calculation and outputting the deviation to the position controller, namely the corresponding rotating speed of each motor, is as follows:
three dimensional column vector gn_attFor normalized attitude gain, a three-dimensional row vector gattFor attitude gain in the parameter file, the rotational inertia of the unmanned aerial vehicle is imn(m, n ═ x, y, z), then
Wherein ixx＝∫∫∫v(y2+z2)ρdV；iyy＝∫∫∫v(x2+z2)ρdV；izz＝∫∫∫v(x2+y2)ρdV；ixy＝∫∫∫v(xy)ρdV；ixz＝∫∫∫v(xz)ρdV；iyz＝∫∫∫v(yz) ρ dV; x, y, z are three directions of coordinate axes, ixx、iyy、izzThe moments of inertia for the x, y, and z axes, ixy、ixz、iyzIs the product of inertia;
three dimensional column vector gn_angFor the normalized angular velocity gain, the angular velocity gain in the parameter file of the parameter server is gangThen there is
Let the allocation matrix be a and,
the inertia matrix I is
The angular acceleration a of the rotor speedrv_angIs provided with
arv_ang＝AT*(A*AT)-1*I； (5)
Let the three-dimensional column vector position error be PerrSpecifying the track position as PctThe actual position is PactThen there is
Perr＝Pact-Pct； (6)
Let the three-dimensional column vector velocity error be verrSpecifying a trajectory velocity vctThe actual speed is vactThen there is
verr＝vact-vct； (7)
Let the unmanned aerial vehicle mass be m, the position gain in the parameter file of the parameter server be gposThe velocity gain is gvelG is gravity acceleration, and a is trajectory accelerationctThen the desired acceleration adesIs composed of
If the elevation angle of the unmanned aerial vehicle is yaw, the specific parameters are as follows:
b3des＝-ades*norm(ades)； (10)
b2des＝b3des× b1des； (11)
wherein, b1des、b3des、b2desTracking a reference trajectory signal, norm (a), for yaw anglesdes) Denotes ades2-norm of (d); the three-dimensional expected positioning matrix R can be obtaineddesIs composed of
Rdes＝[b2des×b3des b2des b3des]； (12)
Let the three-dimensional angle error matrix be AngerrThe positioning matrix is a matrix of R,
the three-dimensional column vector angle error angerrThe method comprises the following specific steps:
wherein, Angerr(a, b) represents AngerrThe row a and the column b of the matrix; let the desired angular rate be ratedes_angThe actual angular rate is rateact_angThen angular rate error rateerr_angIs composed of
Angular acceleration of arateThen there is
arate＝-angerr·gn_att-rateerr_ang·gn_ang+rateact_ang×rateact_ang；
(17)
If the thrust in the z-axis direction is t and the mass of the main unmanned aerial vehicle is m, the
t＝-[0 0 m]*ades； (18)
Then the four-dimensional column vector angular acceleration a of the thrust t is introducedrate_tIs composed of
The final controller obtains the rotor speed omegarotorIs composed of
ωrotor＝arv_ang*arate_t； (20)
arv_angRepresenting the angular acceleration of the rotor speed;
s2.3, each rotor of the main unmanned aerial vehicle receives the control action to make a corresponding response, and the real-time state of each joint changes;
s3, the main unmanned aerial vehicle position controller response module M2 obtains corresponding rotor rotating speed, so that each rotor of the main unmanned aerial vehicle makes a corresponding response to move to a desired position, at the moment, the real-time state of each joint changes, the real-time state parameters of the main unmanned aerial vehicle are transmitted to the main unmanned aerial vehicle deviation operation module M1, the main unmanned aerial vehicle deviation operation module M1 obtains the control function of a deviation output controller through operation and transmits the control function to the main unmanned aerial vehicle position controller response module M2, and a complete negative feedback control loop is formed;
s4, the slave unmanned aerial vehicle following control compensation module M3 based on the ADRC calculates the passive force borne by the slave unmanned aerial vehicle and transferred by the load through an ADRC algorithm, and outputs a ' describing the expected acceleration a ' of the slave unmanned aerial vehicle 'des；
S5, the slave drone 'S position controller response module M4 outputs the slave drone' S rotor speed ω 'by inputting the desired acceleration in the ADRC controller based slave drone following control compensation module M3'rotorAnd the control of the slave unmanned aerial vehicle is realized.
2. The cooperative control method for multiple unmanned aerial vehicles based on ADRC control of claim 1, wherein: the master unmanned aerial vehicle joint state publishing module M0 transmits the real-time state of each joint including coordinates, linear speed and angular speed to the parameter server.
3. The cooperative control method for multiple unmanned aerial vehicles based on ADRC control of claim 1, wherein: in the master unmanned aerial vehicle deviation operation module M1: the node of the controller of the main unmanned aerial vehicle calculates according to the real-time state parameters of the main unmanned aerial vehicle to obtain deviation and outputs the control action of the position controller, namely the corresponding rotating speed of each motor; the real-time state parameters of the main unmanned aerial vehicle comprise the coordinates, linear speed and angular speed of the main unmanned aerial vehicle;
and the main unmanned aerial vehicle position controller response module M2 operates the motors according to the rotating speeds corresponding to the motors output by the main unmanned aerial vehicle deviation operation module M1.
4. The cooperative control method for multiple unmanned aerial vehicles based on ADRC control of claim 1, wherein: step S4 specifically includes the following steps:
s4.1, according to the speed v of the slave unmanned aerial vehicle in the directions of the x axis and the y axisxAnd vyCalculating the magnitude f of the driven force in the directions of the x and y axesxAnd fyThe specific calculation is as follows:
wherein v isx0And vy0Are each t0Velocity v of time-dependent unmanned aerial vehicle in x and y axis directionsx1And vy1Are each t1The speed of the moment in the x and y axis directions, m is the mass of the unmanned aerial vehicle, fxAnd fyAt t for the slave unmanned aerial vehicle respectively0To t1The magnitude of the passive force received in the time in the directions of the x axis and the y axis;
s4.2, converting the f obtained in the step S4.1xAnd fyThe acceleration control quantity u in the x-axis direction and the y-axis direction is obtained by transmitting the acceleration control quantity u into an ADRC controllerxAnd uy；
S4.3, according to the acceleration control quantity u in the step S4.2xAnd uyUpdating a 'desired acceleration of slave unmanned'des。
5. According to claim 4The cooperative control method for the multiple unmanned aerial vehicles based on ADRC control is characterized in that: in step S4.2, the ADRC controller includes a transition process, a nonlinear combination and an extended state observer, b0The method comprises the following steps of roughly estimating control gains of a controlled object motor and a slave unmanned aerial vehicle, namely compensating factors of an extended state observer and disturbance compensation;
the transition process carries out prediction processing on the real-time expected acceleration, and the controlled variable tracks the given input to obtain v1、v2Wherein v is0Is a control purpose, v1Is v0Transition process of v2Is v0Respectively for the difference calculation of the subsequent non-linear combination; in order to track a given input at the fastest speed, a fastest control synthesis function fhan (x) is introduced1，x2R, h), the expression of which is shown in formula (23):
thus, the entire transition process is represented as follows:
wherein sign () is a sign function; r is0The parameter is one of the parameters of the controller, and is specifically adjusted according to the time required by the transition process of the system; h is the system sampling step length, beta01、β02、β03The specific parameter value of (2) is determined by the sampling process;
the extended state observer obtains z by compensating the disturbance of system control, namely the rotor speed input u of the slave unmanned aerial vehicle and the output passive force f of the following control process through an ESO algorithm1And z2And a total disturbance z of the system equivalent to the input side3Wherein z is1And z2For determining the tracking error and its derivative, z3For directly compensating the disturbance, thereby accurately estimating the real-time value of the system operation,and eliminating the influence of disturbance; the specific equation for ESO is as follows:
wherein beta is01、β02And beta03The specific value of the parameter of the controller is determined by the sampling step length h; the fal () function is:
δ () is a unit pulse function and sign () is a sign function; b0Is a compensation factor;
obtaining u by nonlinear combination0I.e. v obtained by the transition element is separately measured1(t) and v2(t) and z by extended state observer1And z2Are subtracted to obtain e1And e2Then e is added1And e2Nonlinear combination is carried out to obtain a control quantity u0The nonlinear combination formula is as follows:
wherein r is a controlled variable gain, and when the controlled variable gain exceeds 100, the influence on the system is greatly reduced, so that the value is not more than 100; h is1For the accuracy factor, increase h1The corresponding overshoot of the output can be reduced, and the adjusting time is increased, and the reciprocal of the adjusting time is similar to the proportional gain; c is a damping factor, increasing c reduces the settling time of the system, i.e., the response is quicker, similar to differential gain; disturbance compensation of final system control, namely, the rotor speed input of the slave unmanned aerial vehicle is in a fixed mode, and the expression is as follows:
the final control amount includes a feedback control amountu0And a compensation control quantity z3。
6. The cooperative control method for multiple unmanned aerial vehicles based on ADRC control of claim 4, wherein: in step S4.3, the desired acceleration a 'of the slave drone is updated'desThe following were used:
in formulae (28) and (29), ux、uyControl quantities a 'of slave unmanned aerial vehicles in x-axis and y-axis directions'desIs the slave drone expected acceleration; u. of0Is controlled by the fastest control synthesis function fhan (x)1，x2R, h) control quantity of output, b0Is the ADRC controller compensation factor, z3Is a state variable of the extended state observer; the updated acceleration a'desAs the expected acceleration of the slave unmanned aerial vehicle following control, and the updated value a 'at that time'desAs a control target of the rotor speed of the slave unmanned aerial vehicle, the real-time stress of the slave unmanned aerial vehicle is adjusted through the motor.
7. The cooperative control method for multiple unmanned aerial vehicles based on ADRC control of claim 1, wherein: in step S5, the desired acceleration M3 is input, and the rotor speed ω 'of the slave drone is output'rotorThe method comprises the following specific steps:
let the elevation angle of the slave unmanned aerial vehicle be yaw', the specific parameters are as follows:
b3′des＝-a′des*norm(a′des)； (31)
b2′des＝b3′des×b1′des； (32)
wherein, b 1'des、b3′des、b2′aesTracking a reference trajectory signal for yaw angle, norm (a'des) Is a'des2-norm of (d); three-dimensional expected positioning matrix R 'can be obtained'desIs composed of
R′des＝[b2′des×b3′des b2′des b3′des]； (33)
Let three-dimensional angle error matrix be Ang'errThe positioning matrix is R', has
Then three-dimensional column vector angular error ang'errIs (Ang'err(a, b) represents Ang'errMatrix a row b column element)
Let desired angular velocity be rat'des_angActual angular rate is rat'act_angThen angular rate error rate'err_angIs composed of
Let the angular acceleration be a'rateThen there is
a′rate＝ang′err·g′n_att-rate′err_ang·g′n_ang+rate′act_ang×rate′act_ang；
(38)
If the thrust in the z-axis direction is t 'and the mass of the slave unmanned aerial vehicle is m', then
t′＝-[0 0 m′]*a′des； (39)
Then four-dimensional column vector angular acceleration a 'of thrust t is introduced'rate_tIs composed of
The final controller obtains the rotor speed omega of the slave unmanned aerial vehicle'rotorIs composed of
ω′rotor＝a′rv_ang*a′rate_t； (41)
a′rv_angRepresenting the rotor speed angular acceleration.
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