Adaptive control method for unmanned vehicle with slung load

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1. An adaptive control method for an unmanned vehicle with a slung load, comprising the steps of:
 establishing a feedback linear controller, such that f(x)=M_{b}(¨_{a}_{d}+Λė_{x}_{a})+C_{b}(&xdot;
_{a}_{d}+Λe_{x}_{a})+D_{b}&xdot;
_{a}+G_{b}, wherein f(x) is a control dynamics input for an unmanned aerial vehicle, x_{a}_{d }is a vector representing roll, pitch, yaw and altitude for the unmanned aerial vehicle based on a corresponding trajectory, M_{b }is an inertia matrix associated with the unmanned aerial vehicle, C_{b }is a centrifugal force and coriolis force matrix associated with the unmanned aerial vehicle, D_{b }is a drag force matrix associated with the unmanned aerial vehicle, and G_{b }is a gravitational vector, &xdot;
_{a }being a velocity vector such that ${\stackrel{.}{x}}_{a}=\left(\begin{array}{c}\stackrel{.}{\phi}\\ \stackrel{.}{\theta}\\ \stackrel{.}{\psi}\\ \stackrel{.}{z}\end{array}\right),$ where φ represents a roll angle of the unmanned aerial vehicle, θ represents a pitch angle of the unmanned aerial vehicle, ψ represents a yaw angle of the unmanned aerial vehicle, and z represents an altitude of the unmanned aerial vehicle, e_{x}_{a }representing control error, and Λ being a positive definite constant matrix such that y=ė_{x}_{a}+Λe_{x}_{a}, where y is a filtered tracking error, and x is a vector defined as $x={\left[{e}_{{x}_{a}}^{T}{\stackrel{.}{e}}_{{x}_{a}}^{T}{x}_{{a}_{d}}^{T}{\stackrel{.}{x}}_{{a}_{d}}^{T}{\ddot{x}}_{{a}_{d}}^{T}\right]}^{T},$ where T is a period of a periodic orbit of the slung load;
establishing a two level neural network such that f(x)=W^{T}σ(V^{T}x)+ε, where W and V are neural network weights, σ represents the sigmoid function, and ε being a known bound;
calculating a neural network estimate of f(x), &fcirc;
(x), as &fcirc;
(x)=Ŵ^{T}σ(&Vcirc;
^{T}x), where Ŵ and &Vcirc;
are actual values of the neural network weights W and V, respectively, given by a tuning algorithm;
transmitting a control input τ for the unmanned aerial vehicle as τ=Ŵ^{T}σ(&Vcirc;
^{T}x)+K_{v}y, where K_{v }is a feedforward gain, for controlling flight of the unmanned aerial vehicle; and
transmitting additional antiswing control input to the unmanned aerial vehicle to correct for a slung load as x_{cor}=K_{xc}Lφ_{L}(t−τ_{xc}) and y_{cor}=K_{yc}Lφ_{L}(t−τ_{yc}), wherein x_{cor }and y_{cor }are additional longitudinal and lateral displacements, respectively, K_{xc }and K_{yc }are, respectively, longitudinal and lateral feedback gains, L is a load cable length, φ_{L }is a load angle in an xz plane, t is time, and τ_{xc }and τ_{yc }are, respectively, longitudinal and lateral time delays introduced in the feedback of the load angle.
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Abstract
The adaptive control method for an unmanned vehicle with a slung load utilizes a feedback linearization controller (FLC) to perform vertical take off, hovering and landing of an unmanned aerial vehicle with a slung load, such as a quadrotor drone or the like. The controller includes a double loop architecture, where the overall controller includes an inner loop having an inner controller which is responsible for controlling the attitude angles and the altitude, and an outer loop having an outer controller responsible for providing the inner loop inner controller with the desired angle values. States, such as including roll, pitch, yaw and/or altitude, are selected as outputs and the feedback linearization technique is used.
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6 Claims
 1. An adaptive control method for an unmanned vehicle with a slung load, comprising the steps of:
 establishing a feedback linear controller, such that f(x)=M_{b}(¨_{a}_{d}+Λė_{x}_{a})+C_{b}(&xdot;
_{a}_{d}+Λe_{x}_{a})+D_{b}&xdot;
_{a}+G_{b}, wherein f(x) is a control dynamics input for an unmanned aerial vehicle, x_{a}_{d }is a vector representing roll, pitch, yaw and altitude for the unmanned aerial vehicle based on a corresponding trajectory, M_{b }is an inertia matrix associated with the unmanned aerial vehicle, C_{b }is a centrifugal force and coriolis force matrix associated with the unmanned aerial vehicle, D_{b }is a drag force matrix associated with the unmanned aerial vehicle, and G_{b }is a gravitational vector, &xdot;
_{a }being a velocity vector such that${\stackrel{.}{x}}_{a}=\left(\begin{array}{c}\stackrel{.}{\phi}\\ \stackrel{.}{\theta}\\ \stackrel{.}{\psi}\\ \stackrel{.}{z}\end{array}\right),$ where φ represents a roll angle of the unmanned aerial vehicle, θ represents a pitch angle of the unmanned aerial vehicle, ψ represents a yaw angle of the unmanned aerial vehicle, and z represents an altitude of the unmanned aerial vehicle, e_{x}_{a }representing control error, and Λ being a positive definite constant matrix such that y=ė_{x}_{a}+Λe_{x}_{a}, where y is a filtered tracking error, and x is a vector defined as$x={\left[{e}_{{x}_{a}}^{T}{\stackrel{.}{e}}_{{x}_{a}}^{T}{x}_{{a}_{d}}^{T}{\stackrel{.}{x}}_{{a}_{d}}^{T}{\ddot{x}}_{{a}_{d}}^{T}\right]}^{T},$ where T is a period of a periodic orbit of the slung load;
establishing a two level neural network such that f(x)=W^{T}σ(V^{T}x)+ε, where W and V are neural network weights, σ represents the sigmoid function, and ε being a known bound;
calculating a neural network estimate of f(x), &fcirc;
(x), as &fcirc;
(x)=Ŵ^{T}σ(&Vcirc;
^{T}x), where Ŵ and &Vcirc;
are actual values of the neural network weights W and V, respectively, given by a tuning algorithm;
transmitting a control input τ for the unmanned aerial vehicle as τ=Ŵ^{T}σ(&Vcirc;
^{T}x)+K_{v}y, where K_{v }is a feedforward gain, for controlling flight of the unmanned aerial vehicle; and
transmitting additional antiswing control input to the unmanned aerial vehicle to correct for a slung load as x_{cor}=K_{xc}Lφ_{L}(t−τ_{xc}) and y_{cor}=K_{yc}Lφ_{L}(t−τ_{yc}), wherein x_{cor }and y_{cor }are additional longitudinal and lateral displacements, respectively, K_{xc }and K_{yc }are, respectively, longitudinal and lateral feedback gains, L is a load cable length, φ_{L }is a load angle in an xz plane, t is time, and τ_{xc }and τ_{yc }are, respectively, longitudinal and lateral time delays introduced in the feedback of the load angle.  View Dependent Claims (2, 3)
 establishing a feedback linear controller, such that f(x)=M_{b}(¨_{a}_{d}+Λė_{x}_{a})+C_{b}(&xdot;
 4. A computer software product that includes a nontransitory storage medium readable by a processor, the nontransitory storage medium having stored thereon a set of instructions for performing adaptive control for an unmanned vehicle with a slung load, the instructions comprising:
 (a) a first set of instructions which, when loaded into main memory and executed by the processor, causes the processor to establish a feedback linear controller, such that f(x)=M_{b}(¨_{a}_{d}+Λė_{x}_{a})+C_{b}(&xdot;
_{a}_{d}+Λe_{x}_{a})+D_{b}&xdot;
_{a}+G_{b}, wherein f(x) is a control dynamics input for an unmanned aerial vehicle, x_{a}_{d }is a vector representing roll, pitch, yaw and altitude for the unmanned aerial vehicle based on a corresponding trajectory, M_{b }is an inertia matrix associated with the unmanned aerial vehicle, C_{b }is a centrifugal force and coriolis force matrix associated with the unmanned aerial vehicle, D_{b }is a drag force matrix associated with the unmanned aerial vehicle, and G_{b }is a gravitational vector, &xdot;
_{a }being a velocity vector such that${\stackrel{.}{x}}_{a}=\left(\begin{array}{c}\stackrel{.}{\phi}\\ \stackrel{.}{\theta}\\ \stackrel{.}{\psi}\\ \stackrel{.}{z}\end{array}\right),$ where φ represents a roll angle of the unmanned aerial vehicle, θ represents a pitch angle of the unmanned aerial vehicle, ψ represents a yaw angle of the unmanned aerial vehicle, and z represents an altitude of the unmanned aerial vehicle, e_{x}_{a }representing control error, and Λ being a positive definite constant matrix such that y=ė_{x}_{a}+Λe_{x}_{a}, where y is a filtered tracking error, and x is a vector defined as$x={\left[{e}_{{x}_{a}}^{T}{\stackrel{.}{e}}_{{x}_{a}}^{T}{x}_{{a}_{d}}^{T}{\stackrel{.}{x}}_{{a}_{d}}^{T}{\ddot{x}}_{{a}_{d}}^{T}\right]}^{T},$ where T is a period of a periodic orbit of the slung load;
(b) a second set of instructions which, when loaded into main memory and executed by the processor, causes the processor to establish a two level neural network such that f(x)=W^{T}σ(V^{T}x)+ε, where W and V are neural network weights, σ represents the sigmoid function, and ε being a known bound;
(c) a third set of instructions which, when loaded into main memory and executed by the processor, causes the processor to calculate a neural network estimate of f(x), &fcirc;
(x), as &fcirc;
(x)=Ŵ^{T}σ(&Vcirc;
^{T}x), where Ŵ and &Vcirc;
are actual values of the neural network weights W and V, respectively, given by a tuning algorithm;
(d) a fourth set of instructions which, when loaded into main memory and executed by the processor, causes the processor to transmit a control input τ for the unmanned aerial vehicle as τ=Ŵ^{T}σ(&Vcirc;
^{T}x)+K_{v}y, where K_{v }is a feedforward gain, for controlling flight of the unmanned aerial vehicle; and
(e) a fifth set of instructions which, when loaded into main memory and executed by the processor, causes the processor to transmit additional antiswing control input to the unmanned aerial vehicle to correct for a slung load as x_{cor}=K_{xc}Lφ_{L}(t−τ_{xc}) and y_{cor}=K_{yc}Lφ_{L}(t−τ_{yc}), wherein x_{cor }and y_{cor }are additional longitudinal and lateral displacements, respectively, K_{xc }and K_{yc }are, respectively, longitudinal and lateral feedback gains, L is a load cable length, τ_{L }is a load angle in an xz plane, t is time, and τ_{xc }and τ_{yc }are, respectively, longitudinal and lateral time delays introduced in the feedback of the load angle.  View Dependent Claims (5, 6)
 (a) a first set of instructions which, when loaded into main memory and executed by the processor, causes the processor to establish a feedback linear controller, such that f(x)=M_{b}(¨_{a}_{d}+Λė_{x}_{a})+C_{b}(&xdot;
1 Specification
1. Field of the Invention
The present invention relates to control for aerial vehicles, and particularly to an adaptive control method for aerial vehicles carrying slung, or suspended, loads. (u, v, w)
2. Description of the Related Art
The position vector R_{H }of the hook with respect to the aerial vehicle 102's center of gravity (e.g.) is given by:R_{H}=x_{H}i_{H}+y_{H}j_{H}+z_{H}k_{H}. (2)The absolute velocity V_{L }of the load is given by:V_{L}=V_{cg}+&Rdot;+Ω×R, (3)where V_{cg }is the absolute velocity of the center of mass of the aerial vehicle 102, R=R_{L}+R_{H }is the position vector of the load with respect to the center of mass of the aerial vehicle 102, and Q=pi_{H}+qj_{H}+rk_{H }is the angular velocity of the aerial vehicle 102. The absolute acceleration a_{L }of the load is:a_{L}=&Vdot;_{L}+Ω×V_{L}. (4)The unit vector in the direction of the gravity force is given by:K_{g}=−sin(θ)i_{H}+sin(Φ)cos(θ)j_{H}+cos(φ)cos(θ)k_{H}. (5)Beside the gravity, there is an aerodynamic force applied on the point mass load. Since the analysis in the background description is restricted to the aerial vehicle 102's motion near hover, the aerodynamics loads on the load will be neglected.
The equations of motion of the load are written by enforcing moment equilibrium about the suspension point, that is, in matrix form:R_{L}×(−m_{L}a_{L}+m_{L}gk_{g})=0. (6)The above equation gives three scalar equations of second order, only the equations in the x and y directions are retained, which represent the equations of motion of the load.
The suspended load introduces additional terms in the rigid body force and moment equations of motion of the aerial vehicle 102, namely load forces and load dynamics. The force and moment loads, F_{HL }and M_{HL}, are shown in
The obtained equations are nonlinear and complicated. For design purposes, these equations are linearized about the hovering conditions. Near hover, the forward speed is nearly zero (i.e., u_{0}=0). Assuming that the aerial vehicle 102's roll angle is also zero, even with the effect of the load on the aerial vehicle 102 (i.e., Φ_{0}=0) simplifies the analysis. At this condition, the load trim equations give the following trim values:θ_{Lo}0 and Φ_{Lo}=−θ_{o}. (9)Imposing the above results to the linearized load equations obtains the following equations of motion for the load:gθ_{L}−g cos(θ_{o})φ+y_{h}&qdot;+&Vdot;+L{umlaut over (θ)}_{L}=0. (10)L{umlaut over (Φ)}_{L}+gΦ_{L}+gθ+(x_{h}−L sin(θ_{o}))cos(θ_{o})&pdot;+z_{h }sin(θ_{o})&rdot;+L cos(θ_{o})sin(θ_{o})&rdot;+cos (θ_{o})&Udot;+sin(θ_{o})&Wdot;=0. (11)The forces exerted by the load on the aerial vehicle 102 are:F_{x}=m_{L}(−g cos[θ_{O}]θ−xh&pdot;+L sin[θ_{o}]&pdot;−&Udot;[t]−L cos[θ_{o}]{umlaut over (Φ)}_{L}),F_{y}=m_{L}(g cos[θ_{o}]Φ−y_{h}&qdot;−&Vdot;−L{umlaut over (θ)}_{L}) andF_{z}=m_{L}(−g sin[θ_{o}]θ−(z_{h}+L cos[θ_{o}])&rdot;−&Wdot;−L sin[θ_{o}]{umlaut over (Φ)}_{L}). (12)The moments in the xyz directions are: φ
These equations are linear and can be formulated in a state space form. If the load state vector is defined as x_{L}=[{dot over (Φ)}_{L }{dot over (θ)}_{L }Φ_{L }θ_{L }]^{T}, the load equations in state space can be written as:E_{L}&xdot;=A_{L}x, (14)where x is the state vector for the load and the aerial vehicle 102 (i.e., x=[x_{H }x_{L}]).Similarly, the effect of the load on the aerial vehicle 102 force terms can be written also as:
The linearized equations of motion of the aerial vehicle 102 and the load can be written in the following state space forms:
Such work, though promising, is not only based on classical control techniques, but is difficult to apply to modem unmanned aerial vehicles, such as quadrotor drones and the like. It would be desirable to provide an antiswing controller for a quadrotor aerial vehicle slung load system near hover flight. Such a controller should be based on timedelayed feedback of the load swing angles. The output from such a controller would be additional displacements that are added to the vehicle trajectory in the longitudinal and lateral directions, for example.
Thus, an adaptive control method for an unmanned vehicle with a slung load addressing the aforementioned problems is desired.
In the present adaptive control method for an unmanned vehicle with a slung load, a feedback linearization controller (FLC) is used to perform vertical take off, hovering and landing of an unmanned vehicle with a slung load, such as a quadrotor drone or the like. To achieve hover condition, the attitude angles and the position have to be stabilized. The outputs of the controller are the set of variables (x, y, z, φ, θ, ψ), where x, y and z are threedimensional Cartesian coordinates, φ represents roll, θ represents pitch, and ψ represents yaw. However, selecting all six outputs makes the system underactuated, as there are only four inputs. Additionally, a coupling problem also exists, thus the output vector in the present method is selected to simply be (φ, θ, ψ, z), where the Cartesian zcoordinate represents altitude. Thus, the present control method involves double loop architecture, where the overall controller includes an inner loop having an inner controller which is responsible for controlling the attitude angles and the altitude, and an outer loop having an outer controller responsible for providing the inner controller of the inner loop with the desired angle values. The inner controller of the inner loop needs to be much faster than the outer controller of the outer loop in order to achieve stability. These four states (φ, θ, ψ, z), or three states (φ, θ, z), are selected as outputs and the feedback linearization technique is used.
As will be described in greater detail below, the inner controller of the inner loop utilizes a two layer neural network. Essentially, as will be described in greater detail below, the overall adaptive control method includes the following steps:
(a) establishing a feedback linear controller, such that f(x)=M_{b}(¨_{a}_{d}+Λė_{x}_{a}) C_{b}(&xdot;_{a}_{d}+Λe_{x}_{a})+D_{b}&xdot;_{a}+G_{b}, where f(x) is a control dynamics input for an unmanned aerial vehicle, x_{a}_{d }is a vector representing roll, pitch, yaw and altitude for the unmanned aerial vehicle based on a corresponding trajectory, M_{b }is an inertia matrix associated with the unmanned aerial vehicle, C_{b }is a centrifugal force and coriolis force matrix associated with the unmanned aerial vehicle, D_{b }is a drag force matrix associated with the unmanned aerial vehicle, and G_{b }is a gravitational vector, &xdot;_{a }being a velocity vector such that
(b) establishing a two level neural network such that f(x)=W^{T}σ(V^{T}x)+ε, where W and V are neural network weights, σ represents the sigmoid function, and ε being a known bound;
(c) calculating a neural network estimate of f(x), &fcirc;(x), as &fcirc;(x)=Ŵ^{T}σ(&Vcirc;^{T}x), where Ŵ and &Vcirc; are actual values of the neural network weights W and V, respectively, given by a tuning algorithm;
(d) providing a control input τ for the unmanned aerial vehicle as τ=Ŵ^{T}σ(&Vcirc;^{T}x)+K_{v}y, where K_{v }is a feedforward gain; and
(e) providing additional antiswing control input to the unmanned aerial vehicle to correct for a slung load as x_{cor}=K_{xc}Lφ_{L}(t−τ_{xc}) and y_{cor}=K_{yc}Lφ_{L}(t−τ_{yc}), wherein x_{cor }and y_{cor }are additional longitudinal and lateral displacements, respectively, K_{xc }and K_{yc }are, respectively, longitudinal and lateral feedback gains, L is a load cable length, φ_{L }is a load angle in an xz plane, t is time, and τ_{xc }and τ_{yc }are, respectively, longitudinal and lateral time delays introduced in the feedback of the load angle.
These and other features of the present invention will become readily apparent upon further review of the following specification.
Unless otherwise indicated, similar reference characters denote corresponding features consistently throughout the attached drawings.
In the present method, a feedback linearization controller (FLC) is used to perform vertical take off, hovering and landing of an unmanned vehicle with a slung load, such as a quadrotor drone or the like. To achieve hover condition, the attitude angles and the position have to be stabilized. The outputs of the controller are the set of variables (x, y, z, φ, θ, ψ), where x, y and z are threedimensional Cartesian coordinates, φ represents roll, θ represents pitch, and ψ represents yaw. However, selecting all six outputs makes the system underactuated, as there are only four inputs. Additionally, a coupling problem also exists, thus the output vector in the present method is selected to simply be (φ, θ, ψ, z), where the Cartesian zcoordinate represents altitude. Additionally, it should be noted that the drag and gyroscopic terms cannot be neglected, and are each considered in order to achieve complete stability. Thus, the present control method involves double loop architecture, as illustrated in
In the following, the position of the body frame of an unmanned aerial vehicle (UAV) 104, such as a quadrotor or the like, with respect to the frame of the Earth is denoted by the vector ξ=[x,y,z]^{T}, and the orientation in angular position of the body frame with respect to the Earth frame is denoted by η=[100 ,θ,ψ]^{T}, which represent pitch, roll and yaw, respectively. The translational and rotational movement of the quadrotor with respect to the Earth inertial frame can be described by using the combined vector ξ and η; i.e., q=[ξ^{T}η^{T}]^{T}.
For the system, the inertia matrix is given by
Further, in what follows, F_{ξ }and τ_{η }represent the translational forces and the torques, respectively. The drag is represented by the matrices D_{η }and D_{ξ}, where k_{t }is the translational drag and k_{r }is the rotational drag, such that
The translational dynamics for the system, discounting the slung load, which will be addressed in detail below, are described by M_{ξ}{umlaut over (ξ)}+G_{ξ}+D_{ξ}{dot over (ξ)}=R_{BI}uA_{1}, where M_{ξ }is a mass term, such that
For ease of understanding, the dynamics may also be represented in the space state format, where p, q and r represent the angular velocities, such that: {dot over (Φ)}=p+q sin φ tan θ+r cos φ tan θ; {dot over (θ)}=q cos φ−r sin φ;
As will be described in greater detail below, the inner loop inner controller 12 utilizes a two layer neural network. Essentially, as will be described in greater detail below, the overall adaptive control method includes the following steps:
(a) establishing a feedback linear controller, such that f(x)=M_{b}(¨_{a}_{d}+Λė_{x}_{a})+C_{b}(&xdot;_{a}_{d}+Λe_{x}_{a})+D_{b}&xdot;_{a}+G_{b}, where f(x) is a control dynamics input for an unmanned aerial vehicle, x_{a}_{d }is a vector representing desired roll, pitch, yaw and altitude for the unmanned aerial vehicle based on a desired corresponding trajectory, M_{b }is an inertia matrix associated with the unmanned aerial vehicle, C_{b }is a centrifugal force and coriolis force matrix associated with the unmanned aerial vehicle, D_{b }is a drag force matrix associated with the unmanned aerial vehicle, and G_{b }is a gravitational vector, &xdot;_{a }being a velocity vector such that
(b) establishing a two level neural network such that f(x)=W^{T}σ(V^{T}x)+ε, where W and V are neural network weights, σ represents the sigmoid function, and ε being a known bound;
(c) calculating a neural network estimate of f(x), &fcirc;(x), as &fcirc;(x)=Ŵ^{T}σ(&Vcirc;^{T}x), where Ŵ and &Vcirc; are actual values of the neural network weights W and V, respectively, given by a tuning algorithm;
(d) providing a control input τ for the unmanned aerial vehicle as τ=Ŵ^{T}σ(&Vcirc;^{T}x)+K_{v}y, where K_{v }is a feedforward gain; and
(e) providing additional antiswing control input to the unmanned aerial vehicle to correct for a slung load as x_{cor}=K_{xc}Lφ_{L}(t−τ_{xc}) and y_{cor}=K_{yc}Lφ_{L}(t−τ_{yc}), wherein x_{cor }and y_{cor }are additional longitudinal and lateral displacements, respectively, K_{xc }and K_{yc }are, respectively, longitudinal and lateral feedback gains, L is a load cable length, φ_{L }is a load angle in an xz plane, t is time, and τ_{xc }and τ_{yc }are, respectively, longitudinal and lateral time delays introduced in the feedback of the load angle.
For the inner loop inner controller 12, a subsystem of the state variables (φ, θ, ψ, p, q, r, z, ż) is considered, where p, q and r are vehicle angular velocities and ż is the time rate of change of altitude. The dynamics of these can be expressed as:
Letting φ_{d}, θ_{d}, ψ_{d }and z_{d }be the desired outputs, denoted as x_{a}_{d}, then the error is defined as:e_{φ}=φ−φ_{d} (23)e_{θ}=θ−θ_{d} (24)e_{ψ}=ψ−ψ_{d} (25)e_{z}=z−z_{d}. (26)The control inputs can be written as:
To estimate the gyroscopic torque term, the propeller speeds must be estimated. These are given by the following relation of the inputs with the propeller speeds:
For the outer loop outer controller 14, the motion of the quadrotor in the horizontal direction is due to the horizontal components of the thrust forces. The roll and pitch angles are important for the horizontal components of the thrust forces and, therefore, to reach a desired position for x and y, desired values of φ_{d }and θ_{d }are to be generated by the outer loop. Further, the inner loop inner controller 12 must be much faster than the outer loop outer controller 14 for the overall control structure to be stable. Considering a subsystem of the states (x, y, &xdot;, &ydot;) and letting φ_{d }and θ_{d }be small angles, then:
A twolayered neural network (NN) is used for function approximation. The advantage of this two layered NN is that it does not require any preselection of a basis set. Further, the restriction of linearity in parameters is overcome. The first layer weights allow the NN to train its own basis set for the system nonlinearities. These weights are tuned by different algorithms, but the algorithm used in the present method is back propagation with a modified tuning algorithm. No preliminary offline training is required in this type of NN and, further, the problem of net weight initialization is not a matter of concern in this approach. The initial weight updates are selected as zero. While the weights are trained online in real time, the proportional derivative (PD) tracking loop carries out its tracking, keeping the error small. When the NN gets trained, the tracking error reduces. The modification in the tuning algorithm is done to improve robustness to disturbances and estimation errors.
The architecture of the neural network used in the present method is show in
The unmanned aerial vehicle (UAV) 104 inner loop inner controller 12 can be described by:M_{b}¨_{a}+C_{b}&xdot;_{a}+D_{b}&xdot;_{a}+G_{b}=τ. (36)The tracking error and the filtered tracking error are defined by e_{x}_{a}=x_{a}_{d}−x_{a}y=ė_{x}_{a}+Λe_{x}_{a}, thus the dynamics, in terms of the filtered error, can be expressed as:M_{b}&ydot;=−C_{b}y+f(x)−τ, (37)where the unknown nonlinear UAV dynamics are defined as:f(x)=M_{b}(¨_{a}_{d}+Λė_{x}_{a})+C_{b}(&xdot;_{a}_{d}+Λe_{x}_{a})+D_{b}&xdot;_{a}+G_{b}. (38)Here, the vector x can be defined as:
It is assumed that on any compact subset of R^{n}, the ideal NN weights are bounded so that:∥Z∥≦Z_{B}, (43)where Z_{B }is known. Letting an NN estimate of f(x) be given byf(x)=Ŵ^{T}σ(&Vcirc;^{T}x) (44)with Ŵ and &Vcirc; being the actual values of the NN weights given by the tuning algorithm, then these are estimates of the ideal weights, and the weight deviation or weight estimation error is defined as:{tilde over (V)}=V−&Vcirc; (45a){tilde over (W)}=W−Ŵ (45b){tilde over (Z)}=Z−&Zcirc;. (45c)The linearity in parameters restriction is overcome by providing tuning algorithms which appear in a nonlinear fashion. The hidden layer output error for a given x is given by:{tilde over (σ)}=σ−{circumflex over (σ)}=σ(V^{T}x)−σ(&Vcirc;^{T}x). (46)The Taylor series expansion of σ(x) for a given x can be written as:σ(V^{T}x)=σ(&Vcirc;^{T}x)+σ′(&Vcirc;^{T}x){tilde over (V)}^{T}x+0({tilde over (V)}^{T}x)^{2}, (47)where
and O(z)^{2 }denotes terms of second order. This equation is very important, since the nonlinear term in {tilde over (V)} is replaced by a linear term and higher order terms. This allows derivation of the tuning algorithm for {tilde over (V)}. The control input is now selected as:τ=Ŵ^{T}σ(&Vcirc;^{T}x)+K_{v}y, (50)where K_{v }is the feedforward gain, which is selected by the designer. The NN weight tuning algorithms are given by:{circumflex over (&Wdot;)}=F{circumflex over (σ)}y^{T }and (51){circumflex over (&Vdot;)}=Gx({circumflex over (σ)}′^{T}Ŵy)^{T}, (52)where the design parameters F and G are positive definite matrices.
Letting the desired trajectory be bounded by a known bound, and the NN weights also bounded by a known assumption and with the weight tuning algorithms provided above with any constant matrices F=F^{T}>0 and G=G^{T}>0, then the filtered error y(t) and the NN weight estimates &Vcirc; and Ŵ are uniformly ultimately bounded. Moreover, the tracking error can be kept very small by proper selection of K_{v}.
The conversion of the NN estimates to the actual angular velocity of the rotor (i.e., the input) is performed as: a) provide the error and reference trajectory and its derivatives; b) provide the weights V and W in the layers of the NN to form f(x); c) update the weights online to give &fcirc;(x), where &fcirc;(x) is the estimate of the dynamics, which is used to obtain the torques τ, which are related to the lifting forces f_{i }of the propellers; and d) the f_{i }values are related to the angular speed.
Timedelayed feedback control, or timedelay autosynchronization, constructs a control value from the difference of the present state of a given system to its delayed value, i.e., s(t)−s(t−τ_{cor}). By proper selection of the time delay τ_{cor}, the control value vanishes if the state to be stabilized converges. Thus, this method is noninvasive. This feedback scheme is easy to implement in an experimental setup and numerical calculation. It is capable of stabilizing fixed points as well as periodic orbits, even if the dynamics are very fast. Also, from a mathematical perspective, it is an appealing instrument, since the corresponding equations fall in the class of delay differential equations.
A very minimal knowledge of the system is required. The only quantity of the system that needs to be known is the period T of the periodic orbit of the slung load, which determines the choice of the time delay. Instead of timedelayed feedback, it is tempting to use proportional feedback, where the control is given by the difference of the current state to the desired state, however this is a complicated process and is numerically exhaustive. Timedelayed feedback has been successfully employed in the context of chaos control, for example.
The timedelayed feedback gives a correction trajectory which adds to the reference trajectory. This correction trajectory is given by the equations:x_{cor}=K_{xc}Lφ_{L}(t−τ_{xc}) and (53)y_{cor}=K_{yc}Lθ_{L}(t−τ_{yc}). (54)These two trajectories act as antitrajectories to the load angles, and these are added to the reference tracking trajectories of the UAV 104. This helps in stabilizing the load angles. The parameters are K_{xc}, K_{yc}, τ_{xc }and τ_{yc}. The feedback gain and the time delay are selected accordingly to stabilize the angles as fast as possible. The optimal values of these can, however, be selected by using optimization algorithms. The values of the gains are selected as K_{xc}=K_{yc}=0.4L, and the time delay is selected as τ_{xc}=τ_{yc}=0.32T_{L}, where T_{L }is given by
With regard to fault tolerant control, a case of failure of one rotor is considered. An assumption is made that a Fault Detection and Isolation (FDI) module is present, which detects and isolates the faults whenever they occur. For example, if rotor #2 fails and there is no lifting force provided by it, then the quadrotor is left with only three rotors to provide the thrust and movement. The rotor on the same axis (which is rotor #4) is controlled until the pitch angle becomes zero. This leaves the quadrotor with only two rotors spinning in one direction, thus making the quadrotor rotate about its vertical axis or in the yaw angle. This is acceptable because the other attitude angles and the altitude are stabilized. The outputs will be (x, y, z, φ, θ). The angle ψ is the yaw angle and it is compromised in the control strategy, so it will not be considered as the output. The control method involves a double loop architecture, where the inner loop inner controller 12 is responsible to control the attitude angles φ and θ and the altitude z, and the outer loop outer controller 14 is responsible for providing the inner loop inner controller 12 with the desired angle values φ_{d }and θ_{d}. Three states (φ, θ, z) are selected as outputs, and feedback linearization is used. Below, the convention ο_{f }represents a fault condition. The input vector is:
It should be understood that the calculations and sets of instructions can be performed by any suitable computer system, such as that diagrammatically shown in
The processor 114 can be associated with, or incorporated into, any suitable type of computing device, for example, a personal computer, a programmable logic controller (PLC) or an application specific integrated circuit (ASIC), for example. The display 118, the processor 114, the memory 112 and any associated computer readable recording media are in communication with one another by any suitable type of data bus, as is well known in the art.
Examples of computerreadable recording media include nontransitory storage media, a magnetic recording apparatus, an optical disk, a magnetooptical disk, and/or a semiconductor memory (for example, RAM, ROM, etc.). Examples of magnetic recording apparatus that can be used in addition to the memory 112, or in place of the memory 112, include a hard disk device (HDD), a flexible disk (FD), and a magnetic tape (MT). Examples of the optical disk include a DVD (Digital Versatile Disc), a DVDRAM, a CDROM (Compact DiscRead Only Memory), and a CDR (Recordable)/RW. It should be understood that nontransitory computerreadable storage media include all computerreadable media, with the sole exception being a transitory, propagating signal.
It is to be understood that the present invention is not limited to the embodiments described above, but encompasses any and all embodiments within the scope of the following claims.