aerial robotics lecture 4_2 nonlinear control

7/25/2019 Aerial Robotics Lecture 4_2 Nonlinear Control

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Lecture 4.2

Nonlinear Control

Up until now, we've considered the problem of controlling the quadrotor at states thatare not too far from the hover equilibrium configuration. Because of this weve been

able to make certain assumptions. We said that the roll and pitch angles were close to

zero and that all the velocities were close to zero.

owever, as we can see from

these pictures, as the vehicle

manoeuvres at high speeds and

e!hibits roll and pitch angles far

awa" from zero, these kinds of

controllers are unlikel" to work.

We want to overcome theselimitations. #pecificall", we want

to consider non$linear controllers

that allow us to control the

vehicle far awa" from the

equilibrium %hover& state.

et's recall the tra(ector"$control

problem. )"picall" we're given a

tra(ector", its derivatives up to the

second order, were specified !%t&,

"%t&, z%t&, and the "aw angle %t&.

&%

&%

&%

&%

&%

t

tz

ty

tx

trT

We're also able to differentiate

them to get higher$order

derivatives.We define an error vector that

describes where the vehicle is,

relative to where it's supposed to

be, and then we tr" to drive that

error e!ponentiall" to zero. B"

insisting on this e!ponential

convergence, we're able to

determine the commanded

accelerations, which then get

translated to inputs that can be

applied b" the quadrotor throughits motors.


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)his is shown in the figure below*

We approached the controller design problem in two stages. #pecificall" there's an

inner loop that addresses the attitude control problem and an outer loop that then

addresses the position control problem*

)he inner loop compares the commanded orientation with the actual orientation and

the angular velocities and determines the input u+. )he outer loop looks at the

specified position and its derivatives, compares that to the actual position and its

derivative, and computes the input u.

We want to do the same thing, e!cept now we want to look at tra(ectories that can

e!hibit high velocities and high roll$ and pitch$angles. )hat is, tra(ectories that are not

close to the equilibrium or hover state.


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We start with a ver" simple control law for the outer position loop*

)his is essentiall" a proportional plus derivative control law. We have the

proportional term, -p, which multiplies the error, er, and the derivative term, -v, that

multiplies the error, re . n addition, there's a feed$forward term,desr which is

essentiall" the acceleration of the specified tra(ector". We're also compensating for

the gravit" vector.

f the s"stem were full" actuated, in other words, if the s"stem could move in an"

direction, this would essentiall" solve the control problem. uwould be a three

dimensional vector and we could control the vehicle, so that the error in the position

would go e!ponentiall" to zero. owever, uis a scalar quantit". n fact, the onl"

direction in which the thrust can be applied is along the b/direction.

Because of that, we do the ne!t$best thing. We take this vector and pro(ect it along the

b/direction.

)hat gives us a pro(ection that is close to the desired input, but not quite, because tisnever perfectl" aligned with b3.


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0e!t, we turn our attention to the attitude$control problem. -nowing that we want t to

be aligned with b/, and knowing that we have ver" little fle!ibilit" in the directions

that t can point in1 we do the ne!t best thing which is to rotate the vehicle, so that b /is

aligned with t.

ttbRdes /

)his gives us the constraint. )he desired rotation matri! should be such that the b /

vector should point in the t direction. We also know what the desired "aw angle needs

to be. 2rmed with these two pieces of information, it's possible to solve for the

desired rotation matri!, Rdes. Using this information, we can now compute the error in

rotation b" comparing the actual rotation, 3, and the desired rotation, 3des*

We ne!t follow the moral equivalent of a proportional$plus$derivative control law*

&%+ eKeKIIu RR

)here are two terms in the parenthesis. 4ne is proportional to the error in rotation, the

other is proportional to the error in angular velocit". We also have to compensate for

the fact that the vehicle has a non$zero inertia, , and there are g"roscopic terms, as we

know from the 5uler equations of motion.

)his gives us the control inputs uand u+. )here are a couple of steps in the derivation

of the control$law that are not straightforward. 6irst, how do we determine the desired

rotation matri!7

We have two pieces of information. 6irst, we want the vector b /to be aligned with the

vector, t. t is known and we want to find 3des, so that this equation is satisfied*

t

tbRdes /

#econd, we want the "aw angle to be equal to the desired "aw angle des. We also

know from the theor" of 5uler$angles that the rotation matri! has the form shownhere*

coscossinsincos

cossincoscossincoscossinsincossincos

sinsincossincossincossinsinsincoscos

R

#o the question is* can we use the two equations shown above to solve for the two

unknown angles, the roll angle and pitch angle.

)he second non trivial step in the derivation of the controller is the calculation of the

error term in the rotation matri!* &% RRe des

R

8iven the current rotation, 3, and the desired rotation, 3des, what is the rotation error7


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We cannot simpl" subtract one matri! from another. f we do, we will get a /!/

matri!, but that matri! is not orthogonal, and therefore is not a rotation matri!.

)he correct wa" to calculate the error is b" asking* what is the magnitude of the

rotation required to go from the current rotation to the desired rotation7

R R

des

)he required rotation is 3, given b"*

desTRRR

We can multipl" 3 b" 3 to verif" that this is, in fact, true.

4nce we have 3, then the angle of rotation which is reall" the magnitude of rotation,

and the a!is of rotation can both be determined using 3odrigues formula.

t turns out that this controller guarantees that the vehicle is stable through a wide

range of angles and velocities. n fact, it's basin of attraction, which is the region from

which it will converge to a desired equilibrium point, is almost all of the space of

rotations, and a ver" large set of angular velocities and linear velocities.

)he inequalities here characterise this basin of attraction. )he first inequalit" specifies

the set of rotations from which it can converge to the hover configuration, and the

second inequalit" tells us the range of angular velocities from which it can converge

to the hover configuration.

0otice that the second inequalit" has a term in the denominator which relates to the

eigenvalues of the inertia tensor. minis essentiall" the smallest eigenvalue of the

inertia matri!. )he smaller this eigenvalue, the larger the quantit" on the right hand

side, which tells us that as we scale down the inertia matri!, the set of angular

velocities from which the robot can converge to the hover state increases. n other

words, the vehicle becomes more stable.

We see this in nature. )here are man" e!amples where the advantage of small size can

be seen. We saw a video showing the abilit" of hone" bees to react to collisions and

recover from them. 2gain, because their inertia is small, the" have a controller whose

base of attraction is fairl" large, and the"'re able to e!ploit the advantages of small

size 9 small inertia to recover from large perturbations. )his is one of the motivations

for us to build smaller quadrotors.

)his picture shows protot"pe of the :ico ;uadrotor, which is onl" cm tip$ to$tip,

weighs onl" +


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We saw a video clip demonstrating the advantage of small size in which two vehicles

were commanded to collide at a relative speed of + ms $. )he" were able to recover

from the collision because of the large basin of attraction for the non$linear controller,

and the vehicles are robust to perturbations like the ones we saw.

n fact, the size of the basin of attraction scales at*

+

>

,?

L

)he smaller the characteristic length, , the larger the basin of attraction.

)he video showed some e!periments that illustrated the advantages of the non$linear

controller over the linear controller. n the videos, the roll and pitch angles deviated

significantl" from the hover position with velocities that scaled up to > ms $.

n one e!periment, the non$linear tra(ector"$controller was used along with two other

controllers*

)he first of these was the linearised hover$controller, and the second was a rotation$

controller or an attitude$controller. B" piecing these three controllers together, the

robot was able to gather momentum before twisting its bod" to go through a window,

whose width was onl" a few centimetres more than the height of the robot.


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)he use of nonlinear controllers also allows us to design more aggressive tra(ectories.

)o see that, it's useful to e!ploit the geometric structure that's inherent in these

quadrotors.

We want to construct two vectors. 4n the right$hand side "ou see the state, q, q , and

the inputs uand u+, but we've now augmented the state 9 inputs with two additional

terms, ,u and ,u . )he vector on the left$hand side consists of the position, velocit",

and acceleration, together with the (erk, (, and snap, s. 2gain, the (erk is the derivative

of the acceleration and the snap the derivative of (erk. is the "aw angle, the rate$

of$change of "aw and is the second$derivative of "aw.

t turns out that there's a one$to$one correspondence between the variables in the

vector on the left and the variables in the vector on the right. 2nd we can see this

from the equations$of$motion*

&//, .% bgamu

/, mju

&% +,/, rpqumsu

,

+

umjp

,

,

u

mjq

r

q

p


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6or e!ample, if we know the acceleration in the z$direction a /, we can calculate uand

b" differentiating that equation, we get an e!pression for ,u . @ifferentiating it again

"ields an e!pression for ,u . 2ll the terms on the right$hand side of these equations are

obtained from the vector consisting of position, velocit", acceleration, (erk, snap, the

"aw angle, its derivative, and its second derivative.

)his shows how we can get the terms in the vector on the right$hand side from the

terms in the vector on the left$hand side. We can go through a similar e!ercise to go

the other wa" around.

What are the implications of this in terms of tra(ector" planning7 What it reall" means

is that if we know the position, the "aw angle, 9 its derivatives, we can determine the

state and the input.

et's consider the equations for a planar quadrotor in which it's easier to visualize the

mapping between these two vectors*

)he equivalence between the two vectors is due to a propert" called differential

flatness. )he ke" idea is that all the state variables and the inputs can be written as

smooth functions of the so$called flat outputs and their derivatives. )he flat outputs

for a quadrotor are simpl" the "$ and z$position. )his equivalence works both wa"s.

+

,

,

,

&%

&%

u

u

u

u

z

y

z

y

z

y

z

y

z

y

z

y

z

y

iv

iv

f we know the state vector, the input u, its first$ and second$derivative, and the input

u+, we can recover the flat outputs and their derivatives*


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)his relationship between the two vectors is called a diffeomorphism. )he term

diffeomorphism refers to a smooth map between these two vectors, and the definition

can be found in most differential$geometr" te!tbooks. #imilarl", for a three$dimensional quadrotor, "ou have a longer vector on both sides, but once again, there's

a one$to$one relationship between these vectors. f we know the flat outputs and their

derivatives, "ou can recover the state variables and the inputs.

)he three$dimensional quadrotor is differentiall" flat, 9 the implication for tra(ector"

planning is quite simple. 3ather than think about the d"namics of the s"stem when we

plan tra(ectories, we can focus our attention on the flat space on the left$hand side. We

can think about mapping obstacles on to this flat space and planning tra(ectories that

are safe, provided that these tra(ectories are smooth.

f the" are smooth, since these are the flat outputs, we're guaranteed to be able to find

feasible state$vectors and inputs that correspond to these tra(ectories. #o once we find

smooth tra(ectories, and, as we've discussed before, we prefer to make these smooth

tra(ectories minimum-snap trajectories, we can then transform them into the real

space with state$vectors and inputs. 2gain, we're guaranteed that these state$vectors

and inputs will satisf" the d"namic equations$of$motion.


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)he video showed an e!ample of a minimum$snap tra(ector" from a starts position to

a goal via an intermediate wa"point. )he tra(ector" was generated in the flat$space,

consisting of the flat$outputs, without specific knowledge of the d"namics, but

because these tra(ectories are smooth, the" could be automaticall" transformed into

tra(ectories that are realistic and can be e!ecuted b" a real robot.

)his reduces the motion$planning problem to a problem in computational geometr". f

we know how to s"nthesize smooth curves going through specified wa"points, a

sub(ect that we've alread" addressed in previous lectures, we can easil" transform

these curves into d"namicall" feasible tra(ectories that a non$linear controller can

faithfull" e!ecute.

aerial robotics lecture 4_2 nonlinear control

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