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By Robert Mahony,Vijay Kumar,and Peter Corke
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SEPTEMBER 20121070-9932/12/$31.00ª2012IEEE
Digital Object Identifier 10.1109/MRA.2012.2206474Date of publication:27August 2012
T
his article provides a tutorial introduction to modeling,es-timation,and control for multi-rotor aerial vehicles that includes the common four-rotor or quad-Aerial robotics is a fast-growing field of robotics and multirotor air-craft,such as the quadrotor (Fig-ure 1),are rapidly growing in popularity.In fact,quadro
tor aerial robotic vehicles have become a standard platform for robotics rearch worldwide.They already have sufficient payload and flight endurance to support a number of indoor and outdoor applications,and the improvements of battery and other technology is rapidly increasing the scope for commercial opportunities.They are highly ma-neuverable and enable safe and low-cost experimentation in mapping,navigation,and control strategies for robots that move in three-dimensional (3-D)space.This ability to move in 3-D space brings new rearch challenges com-pared with the wheeled mobile robots that have driven mobile robotics rearch over the last decade.Small quadrotors have been demon-strated for exploring and mapping 3-D environ-ments;transporting,manipulating,and asmbling objects;and acrobatic tricks such as juggling,balancing,and flips.Additional rotors can be added,leading to general-ized N -rotor vehicles,to improve payload and reliability.
Modeling,Estimation,and Control of Quadrotor
©ISTOCK PHOTO/©ANDREJS ZAVADSKIS
This tutorial describes the fundamentals of the dynamics, estimation,and control for this class of vehicle,with a bias toward electrically powered micro(less than1kg)-scale vehicles.The word helicopt
er is derived from the Greek words for spiral(screw)and wing.From a linguistic perspec-tive,since the prefix quad is Latin,the term quadrotor is more correct than quadcopter and more common than tet-racopter;hence,we u the term quadrotor throughout. Modeling of Multirotor Vehicles
The most common multirotor aerial platform,the quadro-tor vehicle,is a very simple machine.It consists of four individual rotors attached to a rigid cross airframe,as shown in Figure1.Control of a quadrotor is achieved by differential control of the thrust generated by each rotor. Pitch,roll,and heave(total thrust)control is straightfor-ward to conceptualize.As shown in Figure2,rotor i rotates anticlockwi(positive about the z axis)if i is even and clockwi if i is odd.Yaw control is obtained by adjusting the average speed of the clockwi and anticlockwi rotat-ing rotors.The system is underactuated,and the remaining degrees of freedom(DoF)corresponding to the transla-tional velocity in the x-y plane must be controlled through the system dynamics.
Rigid-Body Dynamics of the Airframe
Let f~x,~y,~z g be the three coordinate axis unit vectors without a frame of reference.Let{A}denote a right-hand inertial frame with unit vectors along the axes denoted by f~a1,~a2,~a3g expresd in{A}.One has algebraically that~a1¼~x,~a2¼~y,~a3¼~z in{A}.The vector r¼(x,y,z)2 f A g denotes th
e position of the center of mass of the vehicle. Let{B}be a(right-hand)body fixed frame for the airframe with unit vectors f~b1,~b2,~b3g,where the vectors are the axes of frame{B}with respect to frame{A}.The orientation of the rigid body is given by a rotation matrix A R B¼R¼½~b1,~b2,~b3 2SO(3)in the special orthogonal group. One has~b1¼R~x,~b2¼R~y,~b3¼R~z by construction.
We will u Z-X-Y Euler angles to model this rotation, as shown in Figure3.To get from{A}to{B},we first rotate about a3by the the yaw angle,w,and we will call this inter-mediary frame{E}with a basis f~e1,~e2,~e3g where~e i is expresd with respect to frame{A}.This is followed by a rotation about the x axis in the rotated frame through the roll angle,/,followed by a third pitch rotation about the new y axis through the pitch angle h that results in the body-fixed triad f~b1,~b2,~b3g
R¼
c w c hÀs/s w s hÀc/s w c w s hþc h s/s w
c h s wþc w s/s h c/c w s w s hÀc w c h s/
Àc/s h s/c/c h
@
1
A,
where c and s are shorthand forms for cosine and sine, respectively.
Let v2f A g denote the linear velocity of{B}with respect to{A}expresd in{A}.Let X2f B g denote the angular velocity of{B}with respect to{A};this time expresd in{B}.Let m denote the mass of the rigid object, and I2R333denote the constant inertia matrix(expresd
in the body fixed frame{B}).The rigid body equations of motion of the airframe are[2]and[3]
_n¼v,(1a)
m_v¼mg~a3þRF,(1b)
_R¼R X
3
,(1c)
I_X¼ÀX3I Xþs:(1d) The notation X3denotes the skew-symmetric matrix,such that X3v¼X3v for the vector cross product3and any vector v2R3.The vectors F,s2f B g combine the princi-pal nonconrvative forces and moments applied to the quadrotor airframe by the aerodynamics of the rotors. Dominant Aerodynamics
The aerodynamics of rotors was extensively studied during the mid1900s with the development of manned helicop-ters,and detailed models of rotor aerodynamics are avail-able in the literature[4],[5].Much of the detail about the aerodynamic models is uful for the design of rotor systems,where the whole range of parameters(rotor
x
y
z
{B}
T1
T2
T3
T4
d
Front
Φi
Figure2.Notation for quadrotor equations of motion.N¼4;U i
is a multiple of p=4(adapted with permission from[1]).
Figure1.A quadrotor made by Ascending Technologies with VICON markers for state estimation.
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geometry,profile,hinge mechanism,and much more)are亲子社区
fundamental to the design problem.For a typical robotic quadrotor vehicle,the rotor design is a question for choos-ing one among five or six available rotors from the hobby shop,and most of the complexity of aerodynamic model-ing is best ignored.Nevertheless,a basic level of aerody-namic modeling is required.
The steady-state thrust generated by a hovering rotor (i.e.,a rotor that is not translating horizontally or verti-cally)in free air may be modeled using momentum theory [5,Sec.2.26]as
T i :¼C T q A r i r 2i -2i ,
(2)
where,for rotor i,A r i is the rotor disk area,r i is the radius,
-i is the angular velocity,C T is the thrust coefficient that depends on rotor geometry and profile,and q is the density of air.In practice,a simple lumped parameter model
T i ¼
c T -2i
(3)
is ud,where c T >0is modeled as a constant that can be
easily determined from static thrust tests.Identifying the thrust constant experimentally has the advantage that it will also naturally incorporate the effect of drag on the air-frame induced by the rotor flow.
The reaction torque (due to rotor drag)acting on the airframe generated by a hovering rotor in free air may be modeled as [5,Sec.2.30]
Q i :¼c Q -2i ,
(4)
where the coefficient c Q (which also depends on A r i ,r i ,and
q )can be determined by static thrust tests.
As a first approximation,assume that each rotor thrust is oriented in the z axis of the vehicle,although we note that this assumption does not exactly hold once the rotor begins to rotate and translate through the air,an effect that
is discusd in “Rotor Flapping.”For an N -rotor airframe,we label the rotors i 2f 1ÁÁÁN g in an anticlockwi direc-tion with rotor 1lying on the positive x axis of the vehicle (the front),as shown in Figure 2.Each rotor has associated an angle U i between its airframe support arm and the body-fixed frame x axis,and it is the distance d from the central axis of the vehicle.In addition,r i 2fÀ1,þ1g denotes the direction of rotation of the i th rotor:þ1corre-sponding to clockwi and À1to anticlockwi.The sim-plest configuration is for N even and the rotors distributed symmetrically around the vehicle axis with adjacent rotors counter rotating.
The total thrust at hover (T R )applied to the airframe is the sum of the thrusts from each individual rotor
T R ¼
X N i ¼1
j T i j ¼c T
X
N i ¼1
-2i !:
(5)
The hover thrust is the primary component of the exoge-nous force
F ¼T R ~
z þD (6)
in (1b),where D compris condary aerodynamic forces
that are induced when the assumption that the rotor is in hover is violated.Since F is defined in {B },the direction of application is written ~z ,although in the frame {A }this
direction is ~b 3¼R ~z .
The net moment arising from the aerodynamics (the combination of the produced rotor forces and air resistan-ces)applied to the N -rotor vehicle u are s ¼(s 1,s 2,s 3).
s 1¼c T
X N i ¼1
d i sin (U i )-2i ,s 2¼Àc T X N i ¼1
d i cos (U i )-2i ,
s 3¼c Q
X N i ¼1
r i -2i :
二氧化钒(7)
十八般兵器For a quadrotor,we can write this in matrix form T R
s 1s 2s 3
0B B B B @1
C C C C A ¼c T
c T c T c T
0dc T 0Àdc T Àdc T
dc T
0Àc Q c Q Àc Q c Q
B
B B B @1
C C C C A |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}跳槽简历
C
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B B B B @1
C
C
C C A
,(8)and given the desired thrust and moments,we can solve for the required rotor speeds using the inver of the con-stant matrix C .In order for the vehicle to hover,one must choo suitable -i by inverting C ,such that s ¼0and T R ¼mg .
x , b 1
y , b 2
e 2e 1
X , a 1
Y , a 2
Z, a 3Z, b 3C
O
ψψ
ξ
{A }
{B }
{E }
Figure 3.The vehicle model.The position and orientation of
the robot in the global frame are denoted by n and R ,respectively.
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SEPTEMBER 2012
Blade Flapping and Induced Drag
There are many aerodynamic and gyroscopic effects asso-ciated with any rotor craft that modify the simple force model introduced above.Most of the effects cau only minor perturbations and do not warrant consideration for a robotic system,although they are important for the design of a full-sized rotor craft.Blade flapping and induced drag,however,are fundamental effects that are of significant importance in understanding the natural stabil-ity of quadrotors and how state obrvers operate.The effects are particularly relevant since they induce forces in the x-y rotor plane of the quadrotor,the underactuated directions in the dynamics,that cannot be easily dominated by high gain control.In this ction,we consider a single rotor and we will drop the subscript i ud in the“Dominant Aerodynamics”ction to refer to particular rotors.
Quadrotor vehicles are typically equipped with light-weight,fixed-pitch plastic rotors.Such rotors are not rigid, and the aerodynamic and inertial forces applied to a rotor during flight are quite significant and can cau the rotor to flex.In fact,allowing the rotor to bend is an important property of the mechanical design of a quadrotor and fit-ting rotors that are too rigid can lead to transmission of the aerodynamic forces directly through to the rotor hub and may result in a mechanical failure of the motor mounting or the airframe itlf.Having said this,rotors on small vehicles are significantly more rigid relative to the applied aerodynamic forces than rotors on a full-scale rotor craft.Blade-flapping effects are due to the flexing of rotors,while induced drag is associated primarily with the rigidity of the rotor,and a typical quadrotor will experi-ence both.Luckily,their mathematical expression is equivalent and a single term is sufficient to reprent both effects in a lumped parameter dynamic model.
When a rotor translates laterally through the air it dis-plays an effect known as rotor flapping(e“Rotor Flapping”).A detailed derivation of rotor flapping involves a mechanical model of the bending of the rotor subject to aerodynamic and centripetal forces as it is swept through a full rotation[5,Sec.4.5].The resulting equations of motion are a nonlinear cond-order dynamical system with a dominant highly damped oscillatory respon at the forced frequency corresponding to the an
gular velocity of the rotor.For a typical rotor,the flapping dynamics converge to steady state with one cycle of the rotor[5,p.137],and for the purpos of modeling,only the steady-state respon of the flapping dynamics need be considered.
Assuming that the velocity of the vehicle is directly aligned with the X axis in the inertial frame,v¼(v x,0,0), a simplified solution is given by
b:¼À
l A1c
1À1
2
l2
ÀÁ,b?:¼Àl A1s
1þ1
房子怎么画2
l2
ÀÁ(9)
for positive constants A1c and A1c,and where l:¼j v x j=-r is the advance ,the ratio of magnitude of the horizontal velocity of the rotor to the linear velocity
of rotor tip.The flapping angle b is the steady-state tilt of the rotor away from the incoming apparent wind and b?is the tilt orthogonal to the incident wind.Here,we u equa-tions(4.46)and(4.47)from[5,p.138],noting that adding the effects of a virtual rotor hinge model[5,Sec.4.7]results
in additional pha lag between the sine and cosine com-ponents of the flapping angles[5,Question4.7,p.157]that are absorbed into the constants A1c and A1s in(9).
Rotor flapping is important becau the thrust gener-ated by the rotor is perpendicular to the rotor plane and not to the hub of the rotor.Thus,when the rotor disk tilts the rotor thrust is inclined with respect to the airframe and contains a component in the x and y directions of the body-fixed frame.
In practice the rotors are stiff and oppo the aerody-namic force which is lifting the advancing blade so that its incread thrust due to tip velocity is not fully counteracted by a lower angle of attack and lower lift coefficient—the thrust is incread.Converly for the retreating blade the thrust is reduced.For any airfoil that generates lift(in our ca the rotor blade)there is an associated induced drag
SEPTEMBER2012•IEEE ROBOTICS&AUTOMATION MAGAZINE•23
due to the backward inclination of aerodynamic force with respect to the airfoil motion.The induced drag is proportional to the lift generated by the airfoil.In normal hover conditions for a rotor,this force is equally distrib-uted in all directions around the circumference of the rotor and is responsible for the torque Q(4).However, when there is a thrust imbalance,then the ctor of the rotor travel with high thrust(for the advancing rotor)will generate more induced drag than the ctor where the rotor generates less thrust(for the retreating blade).The net result will be an induced drag that oppos the direc-tion of apparent wind as en by the rotor,and that is proportional to the velocity of the apparent wind.This effect is often negligible for full scale rotor craft,however, it may be quite significant for small quadrotor vehicles with relatively rigid blades.The conquence of blade flapping and induced drag taken together ensures that there is always a noticeable horizontal drag experienced by a quadrotor even when maneuvering at relatively slow speeds.
We will now u the insight from the discussion above to develop a lumped parameter model for exogenous force generation(6).We assume that all four rotors are identical and rotate at similar speeds so that,at least to a first approximation,the flapping respons of the rotors and the unbalanced aerodynamic forces are the same.It follows that the reactive torques on the airframe tran
smitted by the rotor masts due to rotor stiffness cancel.For general motion of the vehicle,the apparent wind results in the advance ratio
l¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v0x2þv0y2
q
=-r,
where v0¼R>v is the linear velocity of the vehicle expresd in the body-fixed frame,with v0x and v0y being the components in the body-fixed x-y plane.Define
A flap¼
1
-R
A1cÀA1s0
A1s A1c0
000
@
1
A,
where-is the t point for the rotor angular velocity. This matrix describes the nsitivities of the flapping angle to the apparent wind in the body-fixed frame,given that l is small and l2is negligible in the denominators of(9). The first row encodes(9)for the velocity along the body-fixed frame x axis.The cond row of A flap is a p=2rotation of this respon to account for the ca where a component of the wind is incoming from the y axis,while the third row projects out velocity in the z axis of the body-fixed frame.
We model the stiffness of the rotor as a simple torsional spring so that the induced drag is directly pr
oportional to this angle and is scaled by the total thrust.The flapping angle is negligible with regard to the orientation of the induced drag,and in the body-fixed frame the induced drag is
D ind:v0%diag(d x,d y,0)v0,
where d x¼d y is the induced drag coefficient.
The exogenous force applied to the rotor can now be modeled by
F:¼T R~zÀT R Dv0,(10) where D¼A flapþdiag(d x,d y,0),and T R is the nominal thrust(5).
An important conquence of blade flapping and induced drag is a natural stability of the horizontal dynam-ics of the quadrotor[7].Define
P h:¼
100
010
(11)
to be the projection matrix onto the x-y plane.The hori-zontal component of a velocity expresd in{A}is
入住证明v h:¼P h v¼(v x,v y)>2R2:(12)
粤菜谱Recalling(1b)and projecting onto the horizontal compo-nent of velocity,one has
m_v h¼ÀT R P h~zþRDv0
ðÞ:
If the vehicle is flying ,v z¼0,then v¼P>h v h and one can write
m_v h¼ÀT R P h~zÀP h RDR>P>h v h,(13)
where the last term introduces damping since,for a typical system,the matrix D is a positive midefinite.
A detailed dynamic model of the quadrotor,including flapping and induced drag,is included in the robotics tool-box for MATLAB[8].This is provided in the form of Simulink library blocks along with a s
原来是这样
et of inertial and aerodynamic parameters for a particular quadrotor.The graphical output of the animation block is shown in Fig-ure4.Simulink models,bad on the blocks,that illus-trate path following and vision-bad stabilization are described in detail in[1].
The discussion provided above does not consider veral additional aerodynamic effects that are impor-tant for high-speed and highly dynamic maneuvers for a quadrotor.In particular,we do not consider transla-tional lift and drag that will effect thrust generation at high speed,axial flow modeling and vortex states that may effect thrust during axial motion,and ground effect that will affect a vehicle flying clo to the ground.It should be noted that high gain control can dominate all condary aerodynamic effects,and high
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