Multi-Rate Path Following Control

8/4/2019 Multi-Rate Path Following Control

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Multi-Rate Path-Following Control for

Unmanned Air Vehicles

D. Antunes C. Silvestre R. Cunha

Instituto Superior Tecnico, Institute for Systems and RoboticsAv. Rovisco Pais, 1, 1049-001 Lisboa, Portugal

{dantunes,cjs,rita}@isr.ist.utl.pt

Abstract: A methodology is provided to tackle the path-following integrated guidance andcontrol problem for unmanned air vehicles with measured outputs available at different rates.The path-following problem is addressed by defining a suitable non-linear path dependenterror space to express the vehicles nonlinear dynamics. The main novelty of the method isto explicitly take into account the different temporal characteristics of the onboard sensor suitein the controller design and implementation. The proposed controller solution relies on a linearparameter varying structure that naturally exploits these multi-rate characteristics of the system

outputs to obtain the desired properties for the resulting integrated guidance and control system.Due to the periodic time-varying characteristics of the multi-rate mechanism, the controllersynthesis is dealt with under the scope of linear periodic systems theory. The effectivenessof the path-following methodology is accessed in simulation for low altitude terrain-followingmaneuvers of a small-scale helicopter using a full dynamic model of the vehicle.

1. INTRODUCTION

This paper presents a path following control solutionfor unmanned air vehicles (UAVs) that naturally takesinto account the multi-rate characteristics of the onboardsensor suite. This feature is of paramount importance inUAVs applications where the linear position measurementsare typically available at a lower rate than that of theremaining variables, as is the case when using the GlobalPositioning System (GPS).

In the field of motion control for autonomous vehicles,path-following has proven to be an efficient alternativeto trajectory tracking. While in the trajectory trackingproblem the vehicle is expected to follow a referencedefined in terms of space and time-coordinates, in thepath-following problem the vehicle is required to convergeand to follow a path without temporal restrictions. In thisway, path-following approaches usually exhibit enhancedperformance when compared to trajectory tracking (?),with smoother convergence to the path and less demandon the control effort. In ? a solution was presented for

the path-following problem for unmanned air vehicles thatrelies on the definition of an error space to accuratelymodel the vehicles nonlinear equations. By taking intoaccount both the kinematics and dynamics this approachprovides an integrated guidance and control solution tothe problem of motion control for UAVs.

Multi-rate control theory has received considerable atten-tion in the last few decades (see, for example ?, ? and the

This work was partially supported by Fundacao para a Cienciae a Tecnologia (ISR/IST pluriannual funding) through the POSConhecimento Program that includes FEDER funds and by thePTDC/EEA-ACR/72853/2006 HELICIM project. The work of D.Antunes was supported by a PhD Student Scholarship, SFRH/BD-

/24632/2005, from the Portuguese FCT POCTI programme.

references therein). If we assume that the sensor samplingand actuators updating rates are related by rational num-bers, a multi-rate system can be modeled as a periodicsystem (?). In applications for autonomous vehicles themulti-rate nature of the sensor suite can either be takeninto account in the navigation system (?) or directly inthe controller. This latter approach is followed in ? wherea gain-scheduling control methodology is proposed for the

control of multi-rate nonlinear systems, which eliminatesthe need to feedforward the values of the state variablesand inputs at trimming. Gain-scheduling control is a pow-erful technique extensively used in aerospace applicationsthat exploits the advantages of linear systems controllerdesign to synthesize a nonlinear compensator, which typi-cally has a linear parameter varying (LPV) structure (?).

In this paper the path-following problem is addressed bydefining a suitable non-linear path dependent error spaceto express the vehicles nonlinear dynamics (see ?). Theproposed controller design methodology relies on a linearparameter varying structure with integral action that takesinto account the multi-rate characteristics of the sensors

and actuators. Based on this structure a gain-schedulingcontroller design and implementation procedure is followed(?), where the multi-rate linear controller synthesis prob-lem is addressed within the scope of the H2 control theoryfor linear discrete time periodic systems.

The proposed technique is applied to the design of a multi-rate integrated guidance and control system for a small-scale helicopter, which is evaluated in simulation for a lowaltitude terrain following maneuver. The reference pathconsidered is composed by the concatenation of straightlines and results from applying a terrain reconstructiontechnique to Laser Range Scanner measurements (?).

The remainder of the paper is organized as follows. We

briefly introduce the helicopter model in Section 2. The


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path-following problem formulation is presented in Sec-tion 3 and the controller design methodology for nonlinearmulti-rate system is described in Section 4. Section 5focuses on the controller design and implementation proce-dure. Finally, simulation results are presented in Section 6and the concluding remarks in Section 7.

The notation is fairly standard. The space ofn-dimensionalcontinuous-time signals, x(t) : R+ Rn, will be denotedby L(R+,Rn) or simply by L(R+) and the space of n-dimensional discrete-time signals, xk : Z

+ Rn, will bedenoted by l(Z+,Rn) or simply by l(Z+). The notationdiag([a1 a2 . . . an]) indicates a block diagonal matrixwhere the entries ai can be either scalar or matrices.Whenever the matrices dimensions are clear, identity andzero matrices are denoted by I and 0. Otherwise thedimensions are explicit indicated as in I3, 032. A vector ofn ones is denoted by 1n = [1 1 . . . 1]

T. Further notation

will be introduced when necessary.

2. HELICOPTER DYNAMIC MODEL

This section summarizes the helicopter dynamic model.For a comprehensive coverage the reader is referred to ?and ? where a full nonlinear dynamic model of a small-scale helicopter is derived from first principles.

Let (IpB, IBR) SE(3) := R3 SO(3) denote theconfiguration of the body frame attached to the vehiclescenter of mass {B} with respect to the inertial frame {I},where I

BR = B

IR T = RZ(B)RY(B)RX(B) is a rotation

matrix parameterized by the Z-Y-X Euler angles =[B B B]

T, B ]/2, /2[, B, B R. In addition,

let v = [uB vB wB]T

and = [pB qB rB]T

, denote thelinear and angular body velocities, respectively, where

v =

B

I R

I

pB, =

B

I R

I

B, and

I

B is the angular velocityof {B} with respect to {I}. The helicopter dynamics canthen be described by using the conventional six degree offreedom rigid body equations of motion

v =1

m[f(v, , u) + B

IR [0 0 g]T] w v

= I1n (v, , u) I1(w Iw)I pB =

I

BR v

= Q(B, B)

, (1)

where m is the vehicle mass, and I is the tensor of inertiaabout the frame {B}. The actuation u = [0 1s 1c 0t]comprises the main rotor collective input 0, the main

rotor cyclic inputs, 1s and 1c, and the tail rotor col-lective input 0t. The force and moment vectors can bedecomposed as f = fmr + ftr + ffus + ftp + ffn andn = nmr + ntr + nfus + ntp + nfn , respectively, wherethe subscripts mr, tr, f us, tp and f n stand for mainrotor, tail rotor, fuselage, horizontal tail plane and verticaltail, respectively. The simulation model includes the rigidbody, main rotor flapping, and Bell-Hiller stabilizing bardynamics, which are not presented here for the sake ofbrevity but can be found in ?.

3. PATH-FOLLOWING FORMULATION

The integrated guidance and control strategy proposed

in ? for the path-following problem consists in defining

a path-dependent transformation, which is applied to thevehicles dynamic and kinematic model to express it ina convenient error space. The problem of steering theunmanned vehicle along a predefined path with a givenvelocity profile, is then reduced to that of regulating theerror variables to zero. In order to present this transforma-tion, we first introduce the frames

{T

}and

{C

}, depicted

in Fig. 1, and a collection of references associated to theseframes. These elements can be briefly described as follows(see ? for further details and derivations):

Frame {T}There is an almost exact correspondence between {T}and the standard Serret-Frenet frame. The x axis, xT,is aligned with the tangent vector to the path, so thatthe linear velocity reference in this frame is given by

vr = Vr [1 0 0]T

where Vr is the desired linear speed. The angular ve-locity reference, also expressed in {T}, can be writtenas

r = Vr [ 0 ]T

where is the torsion and the curvature, whichcharacterize each point on the path. The frame {T}moves along the path attached to the point on thepath closest to the vehicle, meaning that the positionerror can be defined as the two-dimensional vector dt =[dy dz]

T R2 that satisfies0dt

= T

IR(IpB IpT),

where IpT is the position of {T} with respect to {I}.It is also useful to consider the Z-Y-X Euler anglesT = [T T T]

T, which describe the orientation of{T}, and the linear speed VT, which is related to thevehicles velocity by

VT = 11 dy [1 0 0]

T

BR v.

Frame {C}The need to define {C} arises from the fact thatwhile following a path, the vehicle may take differentorientations or even rotate with respect to the path.The reference of orientation for the vehicle is givenby the Z-Y-X Euler angles C = [C C C]

T, C

]/2, /2[ , C, C R. The angular velocity of {C}with respect to {T} expressed in {T} is denoted by TC .The origin of{C} coincides with that of {T}.

Given the definitions of{T} and {C}, the error state vectorxeR11 can be defined as

xe=

veedt

e

=

v BTR vr B

TR (r +

TC)yz

T

IR(IpB IpT) C

, yz =

0 1 00 0 1

.

(2)It is easy to see that the vehicle follows the path with thedesired velocity profile and orientation if and only if xe = 0(regulation problem).

If we restrict the set of possible paths to the set oftrimming paths, the error dynamics can be parameterizedby a set of variables that only depend on the trimmingpath. A trimming path corresponds to a curve that thevehicle can follow while satisfying the trimming conditions,

which is equivalent to having v = 0 , = 0, and u = 0


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{B}{I}

{C}{T}

v

dt

xT

yTzT

vr

IpT

IpB

Fig. 1. Coordinate frames: inertial {I}, body {B}, tangent{T}, and desired body frame {C}

in (1). It is well known that for a vehicle with dynamics

described by (1), the set of trimming trajectories comprisesall z-aligned helices ( = 0, = 0, T = [0 T T]

T, and

T = sign()VT

2 + 2 [ 0 0 1 ]T), followed at constant

speed (VT = 0) and constant orientation with respect tothe path (TC = 0) (?). Consider the following variables:the linear speed reference Vr, the flight path angle T,the yaw orientation of the vehicle with respect to thepath ct = c t and the yaw rate r = Vr

2 + 2.

As discussed in ?, for the case of helicopters this set ofvariables

= (Vr, r, T, ct) (3)

adequately parameterizes the vehicles equilibrium points

corresponding to trimming paths. The error dynamicsdefined with respect to these operating points can bewritten as

P() :=

xe = fe(xe, u , )

ye = he(xe, ), (4)

and is such that fe(0, u, ) = 0 and he(0, ) = 0. Theoutput ye is chosen in such a way that at steady state thecondition ye = 0 implies xe = 0, therefore characterizingan equilibrium point. It can be shown that the outputgiven by

ye =

ve +B

TR

0dt

e

R4 (5)

verifies this condition. By applying integral action to (5),we can guarantee that ye (and consequently xe) goes tozero at steady-state.

Recalling that is a constant parameter vector, thelinearization of P() about (xe = 0, u = u) results ina family of linear time-invariant systems of the form

Pl() =

xe = Ae()xe + Be()uye = Ce()xe

, (6)

where u = u u, Ae() = fexe

(0, u, ), Be() =

fe

u (0, u, ), and Ce() =

he

xe (0, ).

4. NONLINEAR MULTI-RATE CONTROLLERDESIGN

In this section we summarize the gain-scheduling method-ology for nonlinear multi-rate systems presented in ?,which will be applied to the multi-rate path-followingcontrol problem.

4.1 Problem Setup

Consider the nonlinear system

G :=

x(t) = f(x(t), u(t), w(t))

y(t) = h(x(t), w(t))(7)

where x(t) Rn is the state, u(t) Rm is the controlinput, and the vector w(t) Rnw contains referencesr(t) and possibly other exogenous inputs. The vectory(t) Rp can be decomposed as y(t) = [ym(t)T yr(t)T]T =[hm(x(t), w(t))

T hr(x(t), w(t))T]T

where ym(t) Rnym isa vector of measured outputs available for feedback andyr(t)

Rnyr is a vector of tracking outputs, which we

assume to have the same dimensions as the control input,nyr = m. This vector is required to track the reference r(t)with zero steady state error, i.e. the error vector definedas e(t) := yr(t)r(t) must satisfy e(t) = 0 at steady-state.Some of the components of yr(t) may be included in ym(t)as well.

Linearization family

We assume that there exists a unique family of equilibriumpoints for G of the form

:={(x0, u0, w0) : f(x0, u0, w0) = 0, yr0= hr(x0, w0) = r0}which can be parameterized by a vector 0 Rs, suchthat

= {(x0, u0, w0) = a(0), 0 } (8)where a is a continuously differentiable function. We fur-ther assume that there exists a continuously differentiablefunction v such that 0 = v(y0, w0). Applying the functionv to the measured values ofy and w, we obtain the variable

= v(y, w), (9)

which is usually referred to as the scheduling variable.

Linearizing the system G about the equilibrium manifold parameterized by 0 yields the family of linear systems

Gl(0) :=

x(t)y(t)

=

A(0) B1(0) B2(0)

C2(0) D21(0) 0

x(t)w(t)u(t)

(10)

where, e.g. A(0) =fx

(a(0)) and x(t) = x(t) x0.Multi-rate sensors and actuators

We consider that the sample and hold devices that inter-face the discrete-time controller and the continuous-timeplant operate at different rates. The actuators updatingand sensor sampling times do not have to be equallyspaced but the periodicities of the updating and samplingmechanisms are assumed to be rationally related. Letts denote the greatest common divisor of these periodsand consider a set of h-periodic matrices k = k+h,k = k+h taking the form: k = diag(g1(k),...,gp(k)),where gi(k) = 1 if output i is sampled at time tk = kts and

gi(k) = 0 otherwise; k := diag(r1(k),...,rm(k)), where


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rj(k) = 1 if input j is updated at time tk = kts andrj(k) = 0 otherwise. Then the multi-rate input and outputmechanisms, denoted by Smr and Hmr, respectively, canbe modeled as

Smr :L(R+) l(Z+)yk = ymk

yrk = mk 0

0 rk ym(tk)

yr(tk) = ky(tk)

Hmr :l(Z+) L(R+)

k+1 = (I k)k + kuk , 0 = 0uk = (I k)k + kuku(t) = uk t [tk, tk+1[

(11)We also introduce the error variable ek = yrk rk, whererk = rkr(tk).

4.2 Multi-rate controller design and implementation

Consider the linearized system (10) with multi-rate inter-face (11) which can be written as the series connection

SmrGlHmr. Suppose that given a fixed 0 we design alinear controller for this multi-rate system that takes theform depicted in Fig. 2. In the figure CI and CD cor-respond to discrete time linear periodic integrators anddifferentiators, respectively, that can be written as

CI =

xIk+1 = x

I

k + kuI

k

yIk = xI

k + kuI

k

(12)

CD =

xDk+1 = (I mk)xDk + mkuDkyDk = mkxDk + mkuDk

(13)

These two systems are introduced is such a way that thecontroller CK operates in a differential manner, that is, ittakes in the plants output differential values and providesdifferential values to be integrated into the actuation sig-

nals. The importance of this structure for gain-schedulingcontrol will be clarified shortly.

Consider the system Ga shown in Fig. 2, which is givenby the series connection of CI, SmrGlHmr and CD, i.e.Ga = CDSmrGlHmrCI. As proved in ?, this augmentedsystem Ga preserves the detectability and stabilizabilityproperties of the original system Gl, under mild assump-tions. Hence, there exists a stabilizing controller CK, whichis in general periodically time-varying. Furthermore, dueto integral action and given that CK stabilizes the closed-loop system, the structure achieves zero steady-sate errorfor yr. Suppose the equations for CK are given by

CK(0) =xKk+1

yKk

=AKk (0) BK1k(0) BK2k(0)

CKk (0) DK1k(0) DK2k(0)x

K

k

yD

kek

Gl

+

ymk

yrk

rkek

uk

SmrHmr

CICDCK

GaSmrGlHmr

Fig. 2. Regulator structure for non-square systems

and that we have designed a parameter-dependent familyof controllers of the form CICK(0)CD.

Consider then the following implementation for the non-linear gain-scheduled controller, taking the form of a linearparameter varying (LPV) controller and obtained by re-placing the time-frozen parameter 0 with the scheduling

variable k.

K =

xKk+1

yKk

=

AKk (k) B

K

1k(k) BK

2k(k)CKk (k) D

K

1k(k) DK

2k(k)

xKkyDkek

xDk+1yDk

=

I mk mkmk mk

xDk

ymk

xIk+1uk

=

I kI k

xIkyKk

k = v(yk, wk)

xYk+1yk

=

I k kI k k

xYkyk

(14.A)

(14)The scheduling variable k is computed on-line fromthe plant outputs and exogenous variables. The systemdescribed by (14.A) is used to perform a hold operationon the output yk so that the scheduling variable k iscomputed, at each iteration, according to the last sampledvalue of the output. The exogenous vector is assumed tobe available at each sampling instant, so that wk = w(tk).

Since the proposed controller operates in a differentialmanner, we can show that it verifies the following thelinearization property: At each equilibrium point parame-terized by 0, the gain-scheduled controller K linearizesto the designed controller CICK(0)CD. See ? for theimportance of such property and for examples of incorrectimplementations where this property is not verified. Thecontroller structure is inspired in the velocity implemen-

tation (?) - a technique devised to guarantee that thereferred linearization property holds for continuous gain-scheduled controllers, which also operates in a differentialmanner.

5. MULTI-RATE CONTROLLER DESIGN ANDIMPLEMENTATION

We assume that the state variables of the vehicle (1) areall available at a rate of 50 Hz except for the componentsof the linear position, which are assumed to be updated atthe lower rate of 2.5 Hz. The actuation update rate is alsoset to 50 Hz, therefore yielding h = 20 and ts = 0.02.

In accordance with Section 4.1 and assuming that thenonlinear plant is given by (4), we consider the erroroutput yr = ye and define the measured output ym = xe.The set of matrices k and k are determined by

k = I4, mk =

I11 k = 0

diag([13 13 02 13]) otherwise,

rk =

I4 k = 0

diag([03 1]) otherwise, k = 0, . . ,h 1. (15)

Making use of the controller structure presented in Sec-tion 4, a family of linear controllers is designed for theparameterized family of models described by (6) with themulti-rate characteristics just described, using a standard

method of gain-scheduling theory. This method comprises


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the following steps: i) obtain a finite set of parametervalues from the discretization of the continuous parameterspace, ii) synthesize a linear controller for each linear plant(6), obtained from the linearization of the nonlinear plantfor each value of the scheduling parameter, iii) interpolatethe coefficients of the linear controllers to obtain a con-tinuously parameter-varying controller. We detail each of

these steps in the remainder of this section.

5.1 Discretization of the parameter space

The set of parameters that characterize a trimming path,which corresponds to an equilibrium point for the vehiclesdynamic equation, is given by (3). In this paper, werestrict the set of trimming trajectories considered forterrain-following maneuvers to straight-lines (r = 0),with constant linear speed Vr and vehicles yaw anglealigned with the path (ct = 0). The parameter spaceis then reduced to 0 = T. Moreover if we assume thatB = 0 and B = 0 at trimming and consider the variable

= arctan(wBuB )we can show that at equilibrium = 0 (?), and therefore conforms with the general description of schedulingvariable given in Section 4.1.

The selected set of values for this parameter, {0i} is givenby

0 = [-50 -40 -30 -20 -10 0 10 20 30 40 50]

180rad,

which means that the conditions under which the vehicleis expected to operate include following straight lines intrimmed flight, with a flight path angle ranging between-50 and 50 degrees.

5.2 Linear controller synthesis

For linear controller synthesis, the standard LQG solutionfor periodic systems presented in detail in ? was used tosynthesize a controller CK(0i) for each parameter value0i. The augmented system Ga = CDGlCI seen by thecontroller is obtained from (6) and (15). Notice that Gahas nym + nyr = 11 + 4 = 15 outputs and n + m +nyr = 11 +4 +11 = 26 state variables, which determine thedimensions ofCK. The weights defining the LQG problemwere adjusted to yield good transient error responses andsmooth actuation for the closed-loop system.

5.3 Interpolation

The resulting finite set of synthesized controller coeffi-cients, for example {AKk (0i)}, were interpolated usingleast squares yielding a continuously parameter dependentcontroller CICK(0)CD, where the describing matrices arequadratically parameter dependent, for example

AKk (0) = AK1

k + 0AK2

k + 20A

K3

k

BK1k(0) = BK1

1k + 0BK2

1k + 20B

K3

1k .

The disadvantage of this technique is that there is no guar-antee that, even for fixed parameter values, the controllerobtained by interpolation stabilizes the closed loop system.

This analysis was made a posteriori, verifying that for a

Table 1. H2 values vs position sampling rateLinear positionsampling period

0.02 0.04 0.1 0.2 0.4

Closed loop H2 norm 9.69 9.74 9.90 10.15 10.63

dense grid of fixed values of 0 the closed loop system is

stable.Having designed the continuously parameter varying con-troller CICK(0)CD, the final gain-scheduled controllerK takes the form (14), which, as noted earlier, eliminatesthe need to feedforward the trimming values for the statevariables and inputs. Another interesting feature of themethodology is that the implementation of anti-windupschemes is straightforward due to the fact that integralaction is provided at the plants input.

6. RESULTS

The simulation results presented in this section were ob-tained using the non-linear dynamic model SimModHeli,parameterized for the Vario X-Treme model-scale heli-copter (?). The helicopter is required to perform a low alti-tude terrain-following task, by describing the path shownin Figure (3) with constant linear speed Vr = 1.5 m/s.The reference path may result from a terrain followingreconstruction algorithm (see ?) and is divided in the fourfollowing segments: i) a level flight segment along thex axis, ii) a climbing ramp with a flight path angle ofT = 0.5236 rad, iii) a level flight segment along the x axis,and finally iv) a descending ramp with a flight path angleof T = 0.2618 rad. Figure (4) shows the time evolutionof the errors ve = [ue ve we]

T, e = [pe qe re]T, dt, and

e = [e e e]T, actuation, and multi-rate velocity andposition signals. From the figure, we can conclude that thehelicopter with the gain-scheduling multi-rate controllerefficiently performs the desired task. Notice that aftereach transition, the helicopter quickly converges to thereference path corresponding to zero-steady state errors.Furthermore the actuation is kept within the limits ofoperation, exhibiting a smooth behavior.

0 5 10 15 20 25 30 35

-10

-5

0

x[m]

z[m]

TERRAIN

Fig. 3. Helicopter performing a low altitude terrain-following maneuver.

To finalize, for the linear controller synthesis characterizedby 0 = 0, Table 1 shows the loss of performance in termsof the values of the closed-loop H2 norms while changingthe rate of the linear position measurement and keepingthe sampling periods of the other outputs and of the input

at ts = 0.02 s.


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0 5 10 15 201

0.5

0

0.5

1

ms

1

ue

ve

we

0 5 10 15 20

0.50.25

00.250.5

ra

ds

1

pe

qe

re

0 5 10 15 200.2

0.1

0

0.1

0.2

m

dy

dz

0 5 10 15 200.1

0.05

0

0.05

0.1

time [s]

rad

e

e

e

(a) Errors

0 5 10 15 200

0.04

0.08

0

[rad

]

0 5 10 15 20

0

0.01

0.02

1s

[rad]

0 5 10 15 20

0.025

0

0.025

1c

[rad]

0 5 10 15 200

0.05

0.1

0t

[rad]

time [s]

(b) Actuation

0 5 10 15 201

0

1

2

ms

1

uB

vB

wB

0 5 10 15 2020

0

20

40

time [s]

m

xB

yB

zB

(c) Velocity and Position measurements

Fig. 4. Time evolution of the vehicle variables.

7. CONCLUSIONS

In this paper, the path-following problem for unmanned

air vehicles was tackled using an integrated guidance and

control approach. The solution consists in reducing thepath-following problem to that of regulating an adequatelydefined error vector to zero. To this end, a gain-schedulingcontrol methodology was presented that takes into accountthe multi-rate characteristics of the measured outputs.Simulation results showed good performance of the result-ing multi-rate integrated guidance and control system in a

low altitude terrain following task, when the linear positionis available at a rate lower than that of the remainingvariables.

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R. Cunha and C. Silvestre. Dynamic modeling andstability analysis of model-scale helicopters with bell-hiller stabilizing bar. In AIAA Guidance, Navigation,and Control Conference, TX. AIAA.

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