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Page 1: Rotorcraft UAVs at PoliMI

PO

LIPO

LI

di

di M

IM

Itecn

ico

tecn

ico

lano

lano

ADAPTIVE AUGMENTED CONTROL OF UNMANNED ROTORCRAFT VEHICLES

C.L. Bottasso, R. Nicastro, L. Riviello, B. Savini

Politecnico di Milano

AHS International Specialists' Meeting onUnmanned Rotorcraft

Chandler, AZ, January 23-25, 2007

ADAPTIVE AUGMENTED CONTROL OF UNMANNED ROTORCRAFT VEHICLES

C.L. Bottasso, R. Nicastro, L. Riviello, B. Savini

Politecnico di Milano

AHS International Specialists' Meeting onUnmanned Rotorcraft

Chandler, AZ, January 23-25, 2007

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POLITECNICO di MILANO DIA

Rotorcraft UAVs at PoliMIRotorcraft UAVs at PoliMI

• Low-cost platform for development and testing of navigationnavigation and controlcontrol strategies (including vision, flight envelope protection, etc.);

• Vehicles: off-the-shelf hobby helicopters;

• On-board control hardware based on PC-104 standard;

• Bottom-up approach: everything is in-house developedeverything is in-house developed (Inertial Navigation System, Guidance and Control algorithms, Linux-based real-time OS, flight simulators, etc. etc.)

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POLITECNICO di MILANO DIA

OutlineOutline

• Non-linear model predictive control;

• Reference Augmented Predictive Control (RAPC): motivations;

• Reference Augmented Model Identification;

• Reference Augmented Neural Control;

• Results;

• Conclusions and outlook.

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POLITECNICO di MILANO DIA

UAV Control ArchitectureUAV Control Architecture

Target

Obstacles

Hierarchical three-layer control architectureHierarchical three-layer control architecture (Gat 1998):

Vision/sensor range

• Strategic layer: assign mission objectives (typically relegated to a human operator);

•Tactical layer: generate vehicle guidance information, based on input from strategic layer and sensor information;• Reflexive layer: track trajectory generated by tactical layer, control, stabilize and regulate vehicle. In this paper: Adaptive Non-linear Model Predictive Adaptive Non-linear Model Predictive ControlControl.

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POLITECNICO di MILANO DIA

Non-Linear Model Predictive Control

Non-Linear Model Predictive Control

Non-linear Model Predictive ControlNon-linear Model Predictive Control (NMPC):

Find the control action which minimizes an index of performance, by predicting the future behavior of the plant using a non-linear reduced modelnon-linear reduced model.

- Reduced model:

- Initial conditions:

- Output definition:

Cost:

with desired goal outputs and controls.

Stability resultsStability results: Findeisen et al. 2003, Grimm et al. 2005.

L(y;u) = (y ¡ y¤)T Q(y ¡ y¤) + (u ¡ u¤)T R (u ¡ u¤)

minu ;x ;y

J =

Z t0+Tp

t0

L(y;u)dt

s.t.: f ( _x;x;u) = 0 t 2 [t0;t0 +Tp]

x(t0) = x0

y = g(x) t 2 [t0;t0 + Tp]

(¢)¤

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POLITECNICO di MILANO DIA

Model-Adaptive Predictive ControlModel-Adaptive Predictive Control

1. Tracking problem

Plant response

3. Reduced model update

Predictive solutions

2. Steering problem

Prediction window

Steering window

Tracking cost

Prediction error

Prediction window

Tracking cost

Steering window

Prediction error

Tracking costPrediction window

Steering window

Prediction error

Goal trajectory

Receding horizon controlReceding horizon control:

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POLITECNICO di MILANO DIA

MotivationMotivation

• For any given problem: wealth of knowledgeknowledge and legacylegacy methods which perform reasonably well;

• Quest for better performance/improved capabilities: undesirableundesirable and wastefulwasteful to neglect valuable existing knowledge;

Reference Augmented Predictive ControlReference Augmented Predictive Control (RAPCRAPC): exploit available legacy methods, embedding them in a non-linear model predictive control framework.

Specifically:

• ModelModel: augment flight mechanics rotorcraft models (BEM+inflow theories) to account for unresolved or unmodeled physics;

• ControlControl: design a non-linear controller augmenting linear ones (LQR) which are known to provide a minimum level of performance about certain linearized operating conditions.

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POLITECNICO di MILANO DIA

OutlineOutline

• Non-linear model predictive control;

• Reference Augmented Predictive Control (RAPC): motivations;

• Reference Augmented Model Identification;

• Reference Augmented Neural Control;

• Results;

• Conclusions and outlook.

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POLITECNICO di MILANO DIA

GoalGoal:

• Develop reduced modelreduced model capable of predicting the behavior predicting the behavior of the plantof the plant with minimum error (same outputs when subjected to same inputs);

• Reduced model must be self-adaptiveself-adaptive (capable of learning) to adjust to varying operating conditions.

Predictive solutions

Prediction (tracking) window

Steering window

Prediction Prediction error to be error to be minimizedminimized

Reference Augmented Model Identification

Reference Augmented Model Identification

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POLITECNICO di MILANO DIA

Neural augmented reference modelNeural augmented reference model:

reference (problem dependent) analytical model,

RemarkRemark: reference model will notnot, in general, ensure adequateadequate predictions, i.e.

when = system states/controls,

= model states/controls.

Augmented reference model:Augmented reference model:

where is the unknownunknown reference model defectdefect that ensures

when

Hence, if we knewif we knew , we would have perfect predictionperfect prediction capabilities.

d

d

eu = u.

eu = u;

f ref( _x;x;u) = 0:

ex 6= x

x;u

ex; eu

f ref( _x;x;u) = d(x;u);

ex = x

Reference Augmented Model Identification

Reference Augmented Model Identification

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POLITECNICO di MILANO DIA

Approximate with single-hidden-layer neural networkssingle-hidden-layer neural networks:

where

and

= functional reconstruction error;

= matrices of synaptic weights and biases;

= sigmoid activation functions;

= network input.

The reduced model parametersreduced model parameters

are identified on-line using Kalman filteringKalman filtering.

d

¾(Á) = (¾(Á1); : : : ;¾(ÁN n))T

d(y;u) = dp(x;u;pm) +";

dp(x;u;pm) = WmT ¾(Vm

T i +am) + bm;

"

Wm;Vm;am;bm

i = (xT ;uT )T

pm = (::: ;Wmi k;Vmi k

;ami;bmi

; : : :)T

Reference Augmented Model Identification

Reference Augmented Model Identification

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POLITECNICO di MILANO DIA

Model Augmentation ResultsModel Augmentation ResultsPitch rate for plant, reference, and neural-augmented reference with same prescribed inputs.

Short Short transient = transient =

fast adaptionfast adaption

Black: plant

Red: reference model

Blue: reference model +neural network

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POLITECNICO di MILANO DIA

OutlineOutline

• Non-linear model predictive control;

• Reference Augmented Predictive Control (RAPC): motivations;

• Reference Augmented Model Identification;

• Reference Augmented Neural Control;

• Results;

• Conclusions and outlook.

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POLITECNICO di MILANO DIA

Prediction problem:

Enforcing optimalityEnforcing optimality, we get:

Non-Linear Model Predictive Control

Non-Linear Model Predictive Controlminu ;x ;y

J =

Z t0+Tp

t0

L(y;u)dt

s.t.: f ( _x;x;u) = 0 t 2 [t0;t0 +Tp]

x(t0) = x0

y = g(x) t 2 [t0;t0 + Tp]

f ( _x;x;u;pm) = 0; t 2 [t0;t0 +Tp];

x(t0) = x0;

¡d(f T

; _x ¸ )

dt+ f T

;x ¸ + yT;x L ;y = 0; t 2 [t0;t0 +Tp];

¸ (t0 +Tp) = 0;

L ;u +f T;u ¸ = 0; t 2 [t0;t0 +Tp]:

• Model equations:

• Adjoint equations:

• Transversality conditions:

• State initial conditions:

• Co-state final conditions:

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POLITECNICO di MILANO DIA

FuturePast

Prediction window

x0

t0

t0 t0 +Tp

Goal response x¤(t)

Goal control u¤(t)

Non-Linear Model Predictive Control

Non-Linear Model Predictive Control

u(t)

Optimal control u(t)

x(t); t < t0

u(t); t < t0

Â(¢;¢;¢;¢)

It can be shown that minimizing minimizing controlcontrol is See paper for details.

u(t) = ¡x0;y¤(t);u¤(t);t

¢

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POLITECNICO di MILANO DIA

Reference augmented form:Reference augmented form:

where is the unknown control defect.

RemarkRemark: if one knew , the optimal control would be available without having to solve the open-loop optimal control problem.

IdeaIdea:

- ApproximateApproximate using an adaptive parametric element:

- IdentifyIdentify on-line, i.e. find the parameters which minimize the reconstruction error .

pc"

Reference Augmented Predictive Control

Reference Augmented Predictive Control

u(t) = uref(t) +À¡x0;y¤(t);u¤(t);t

¢

À(¢;¢;¢;¢)

À(¢;¢;¢;¢)

À(¢;¢;¢;¢)À

¡x0;y¤(t);u¤(t);t

¢= Àp

¡x0;y¤(t);u¤(t);t;pc

¢+"c

Àp(¢;¢;¢;¢)

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POLITECNICO di MILANO DIA

Iterative procedureIterative procedure to solve the problem in real-time:

• Integrate reduced model equations forwardforward in time over the prediction window, using and the latest available parameters (state prediction):(state prediction):

• Integrate adjoint equations backward backward in time (co-state (co-state prediction):prediction):

• CorrectCorrect control law parameters , e.g. using steepest descent:

pc

uref pc

_pc = ¡ ´ J ;pc ! pnewc = pold

c ¡ ´ J ;pc

¡d(f T

; _x ¸ )

dt+ (f ;x +uT

;x f ;u )T ¸ + yT;x L ;y + uT

;x L ;u = 0 t 2 [t0;t0 + Tp]

¸ (t0 + Tp) = 0

f ( _x;x;u;pm) = 0 t 2 [t0;t0 + Tp]

x(t0) = x0

On-line Identification of Control Parameters

On-line Identification of Control Parameters

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POLITECNICO di MILANO DIA

RemarkRemark: the parameter correction step

seeks to enforce the transversality conditiontransversality condition

Once this is satisfied, the control is optimaloptimal, since the state and co-state equations and the boundary conditions are satisfiedsatisfied.

_pc = ¡ ´ J ;pc

Z t0+Tp

t0

ÀT;pc

(L ;u + f T;u ¸ ) dt = 0

On-line Identification of Control Parameters

On-line Identification of Control Parameters

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POLITECNICO di MILANO DIA

Tracking cost

Future Target

• PredictPredict state forward• PredictPredict co-state backwards

• PredictPredict control action

• UpdateUpdate estimate of control action, based on transversality violation

_pc = ¡ ´ J ;pc

• AdvanceAdvance plant• UpdateUpdate model, based on prediction error

Past

Optimal control

Prediction error

• RepeatRepeat

FuturePast

Prediction horizonSteering window

State

Control

On-line Identification of Control Parameters

On-line Identification of Control Parameters

x(t)

¸ (t)

u(t)

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POLITECNICO di MILANO DIA

Neural-Network-Based Implementation

Neural-Network-Based Implementation

- Drop dependence on time history of goal quantities:

- Approximate temporal dependence using shape functions:

- Associate each nodal value with the output of a single-single-hidden-layerhidden-layer feed-forward neural networkneural network, one for each component:

where

Output:

Input:

Control parameters:

Àp¡x0;y¤(t);u¤(t);t;pc

¢¼Àp

¡x0;y¤(t0);u¤(t0);t;pc

¢

Àp¡x0;y¤(t0);u¤(t0);¿;pc

¢¼

(1¡ ») Àpk

¡x0;y¤(t0);u¤(t0);pc

¢+»Àpk + 1

¡x0;y¤(t0);u¤(t0);pc

¢

oc = W Tc ¾(V T

c i c +ac) +bc

oc = (ÀTp0

;ÀTp1

; : : : ;ÀTpM ¡ 1

)T

i c =¡xT

0 ;x¤T (t0);u¤T (t0)¢T

pc = (::: ;Wci j; : : : ;Vci j

; : : : ;aci; : : : ;bci

; : : :)T

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POLITECNICO di MILANO DIA

FuturePast

Prediction window

x0

t0

t0 t0 +Tp

x(t); t < t0

u(t); t < t0

u¤(t0)

x¤(t0)

NNÀpk

Neural-Network-Based Implementation

Neural-Network-Based Implementation

x¤(t)

u¤(t)

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POLITECNICO di MILANO DIA

OutlineOutline

• Non-linear model predictive control;

• Reference Augmented Predictive Control (RAPC): motivations;

• Reference Augmented Model Identification;

• Reference Augmented Neural Control;

• Results;

• Conclusions and outlook.

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POLITECNICO di MILANO DIA

Vehicle Model and Simulation Environment

Vehicle Model and Simulation Environment

Vehicle modelVehicle model:

• Blade element and inflow theory (Prouty, Peters);

• Quasi-steady flapping dynamics, aerodynamic damping correction;

• Look-up tables for aerodynamic coefficients of lifting surfaces;

• Effects of compressibility and downwash at the tail due to main rotor;

• Process and measurement noise, delays.

Reflexive controllerReflexive controller:

• State reconstruction by Extended Kalman Filtering;

• Reference controller: output-feedback LQR at 50 Hz;

• Goal trajectory planned as in Bottasso et al. 2007.

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POLITECNICO di MILANO DIA

ResultsResults

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ResultsResults

Integral tracking error vs. length of prediction window:

Significant Significant improvement over improvement over LQRLQR

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ResultsResults

Turn rate vs. time:

RAPC

LQR

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ResultsResults

Integral tracking error vs. model mismatch parameter:

RAPC without model adaption

RAPC with model adaption

Significant Significant improvement over improvement over LQRLQR

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ResultsResults

Main rotor collective & norm of control network parameters:

Initial Initial transienttransient

AdaptedAdapted

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ConclusionsConclusions

• Non-linear reduced model identificationreduced model identification for capturing unmodeled or unresolved physics;

• Linear controller promotedpromoted to non-linear;

• Hard real-timereal-time capable (fixed number of ops, no iterations);

• Adaption of control action can be performed independentlyindependently from adaption of reduced model;

• Reference model and reference control ensure good predictions even before adaptionbefore adaption, avoid need for pre-training, simplify adaptionsimplify adaption since defect is small;

• Conceptually possible (but not investigated here) to do adaption diagnosticsadaption diagnostics by monitoring defects;

• Theoretically non-linearly stablenon-linearly stable (if identification of , successful);

• Basic concept demonstrated in a high-fidelity virtual environment.

pcpm

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OutlookOutlook

• Real-time implementation and integration in a rotorcraft UAV (in progress) at the Autonomous Flight Lab at PoliMI;

• Testing and extensive experimentation;

• Integration with vision for fully autonomous navigation in complex environments.