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=

-v_i, & |v_i| > V_M

_m (_i),& _i0, |v_i| V_M

_M (_i),& _i > 0, |v_i| V_M

for \(i=1,2,3\). The values of the constants \(V_m,\alpha_m,\alpha_M\) should be tuned according to bounds on the functions \(f_3,f_4,g_3\); however, in practice these values are tuned from simulation results based on the efficiency of the tracking performance. Once the pseudo-inputs \(v_i\) are computed, they can be transformed back to the actual input signals \([\delta_e,\delta_a,\delta_r]\) through multiplication by \(g_3(V)\). The HOSM design removes the chattering from input signals while increasing robustness to unmodeled dynamics by enforcing the second-order HOSM condition \(\lambda_i = \dot{\lambda_i} = 0\).

&=& _k=0^n

&=& _k=0^n n k p^k (1-p)^n-k (1 - e^-k )

&=& _k=0^n n k p^k (1-p)^n-k

&-& _k=0^n n k (p e^- )^k (1-p)^n-k

&=& 1 - ^n.

cc _0 & =

_0=

Let \(\mathcal{S}(t)= S{\left[\mathbf{q}(t),\:\mathbf{p}(t)\right]}\) be the 3D vehicle position at time \(t\) with \(\mathbf{q}(t)=\left[q_{0}\: q_{1\:}q_{2\:}q_{3}\right]^{T}\) and \(\mathbf{p}(t)=\left[p_{0}\: p_{1}\: p_{2}\right]^{T}\) representing the attitude as a unit quaternion and the vehicle position in meters. \(\mathcal{S}(t)\) can be computed as follows:

m &= -T R e_3+ F_e(,,R,,,d(t)) + R _R \label{psys2}

&= R() \label{psys3}

&= -()+ +_e(,,R,,,d(t))+_g + _TTe_3 \label{psys4}

\(m\) is the vehicle’s mass and \(\mathbf{I} \in{{\mathbb R}}^{3\times 3}\) is its inertia matrix.

\(\cal{I}\)=\(\{O;\vec{\imath}_o,\vec{\jmath}_o,\vec k_o\}\) is a fixed (inertial or Galilean) frame with respect to which the vehicle’s absolute pose (position + orientation) is measured. This frame is typically chosen as the NED frame (North-East-Down) with \(\vec{\imath}_o\) pointing to the North, \(\vec{\jmath}_o\) pointing to the East, and \(\vec k_o\) pointing to the center of the earth. \(\cal B\)=\(\{G;\vec{\imath},\vec{\jmath},\vec k\}\) is a frame attached to the body. The vector \(\vec k\) is parallel to the thrust force axis. This leaves two possible and opposite directions for this vector. The direction chosen here (with \(\vec k\) pointing downward nominally) is consistent with the convention used for VTOL vehicles (see Fig. \ref{fig1}).

\(\xi=(\xi_1,\xi_2,\xi_3)^\top \in {{\mathbb R}}^3\) is the vector of coordinates of the vehicle’s CoM position expressed in the inertial frame \({\cal I}\).

\(R \in SO(3)\) is the rotation matrix representing the orientation of the body-fixed frame \(\mathcal{B}\) with respect to the inertial frame \(\mathcal{I}\). The column vectors of \(R\) correspond to the vectors of coordinates of \(\vec{\imath},\;\vec{\jmath},\;\vec k\) expressed in the basis of \(\cal I\).

\(\omega=(\omega_1,\omega_2,\omega_3)^\top \in {{\mathbb R}}^3\) is the angular velocity vector of the body-fixed frame \(\cal B\) relative to the inertial frame \(\cal I\) and expressed in \(\cal B\).

\(\Gamma_e\) is the external torque vector induced by all external forces.

\(\mathrm{S}(\cdot)\) is the skew-symmetric matrix associated with the cross product (*i.e.*, \(\mathrm{S}(u)v = u \times v,\forall u,v\in{{\mathbb R}}^3\)).

\(d(t)\) represents external disturbances, including wind effect, which do not depend on the vehicle’s position and motion.

\(\tau_g\) is the gyroscopic torque associated with rotor crafts.

\(e_3=(0,0,1)^\top\) is the third vector of the canonical basis of \({{\mathbb R}}^3\) and also the vector of coordinates in \(\cal B\) of the thrust direction vector \(\vec k\).

\(\Sigma_{T}\) and \(\Sigma_R\) denote \(3 \times 3\) (approximately) constant coupling matrices.

_R =

0 & _1&0

_2&_3&_4

0 & 0&0

_R=-(e_3),

m &= -T R e_3+ F_e(, d(t)) + R _R \label{sys2}

&= R() \label{sys3}

&= -()+ \label{sys4}

m & = -mg()e_3 -Te_3 +_R \label{sy2}

& = \label{sy3}

& = \label{sy4}

m_3&= -T

_3 &= _3

u= -k_0 x_1 -k_1 x_2-k_2 x_3-k_3 x_4, k_i>0,

p^4+(k_3+k_1)p^3 +(k_2+k_0) p^2 + k_1 p + k_0,

to yield the exponential stabilization of the origin of the system(1)–(s4). In this case, when \(\varepsilon\) is small the term \(\varepsilon u\) affects the control performance marginally only. In fact, it suffices to choose the control gains \(k_i\) (\(i=0, \ldots, 4\)) so that the characteristic polynomial of the closed-loop system, given by

X&=(X_1,X_2,X_3,X_4)^

&=(,m,-T R e_3+mge_3,-R)^,

&= (T_2, -T_1,)^,

U&= R ( -T e_3 + T (e_3) ^-1+2 T(e_3) -T ()^2 e_3

&- T(e_3)^-1 () ).

with

\[\dot{\cal V}= -k_0\frac{| \eta \times \eta_d|^2}{(1\!+\! \eta^\top \eta_d)^2}.\]

(T, _d) = (||F(,t)|, )

c z = k_z (- z + sat__z(z+)), |z(0)| < _z,

(**x) = _i=1^n _j_i ( x_i-x_j-d)^2.**

\[1_S(\bf s) = \left\{ \begin{array}{lcl} 1 & \hbox{if} & \bf s\in S \cr 0 & \hbox{if} & \bf s\notin S \cr \end{array} \right.\]

^(m-n/2 + _n + ^(m-n)/2

A B C D E

A = B^2

( ^n_ j=1 _j ) H_c= _ij (i|i)

Formally, a *block feedforward system* is defined by \(r\) blocks such that

&_1=f_1(x_1,u), & \label{bf1}

&_2=f_2(x_1,x_2,u),&

&&

&_r=f_r(x_1,x_2,…,x_r,u),& \label{bf2}

where \(x_{i}\in R^{n_{i}}\) and \(u\) is the vector of inputs.

x_1 x_2+x_1^2 x_2^2 + x_3,

x_1 x_3+x_1^2 x_3^2 + x_2,

x_1 x_2 x_3.

(x_1 x_2 x_3 x_4 x_5 x_6)^2

+ (y_1 y_2 y_3 y_4 y_5 + (y_1 y_3 y_4 y_5 y_6 + (y_1 y_2 y_4 y_5 y_6 + (y_1 y_2 y_3 y_5 y_6)^2

+ (z_1 z_2 z_3 z_4 z_5 + (z_1 y_3 z_4 z_5 z_6 + (z_1 y_2 z_4 z_5 z_6 + (z_1 y_2 z_3 z_5 z_6)^2

+(u_1 u_2 u_3 u_4 + u_1 u_2 u_3 u_5 +u_1 u_2 u_4 u_5 + u_1 u_3 u_4 u_5)^2

x_1 + y_1 + ( _i < 5 + a^2 )^2

h(x) &= ( + ) dx

&= dx -2 (x-2)

There are 33 examples at this point.

——————————————————————–

——————————————————————–

x&= 3 + +

y&= 4 +

z &= 5 +

u &= 6 +

= 5 + a +

= 12

= 13

= 11 + d

(

1 & 0 & …& 0

0 & & 1 & …& 0

& & &

0 & 0 & …& 1

]

r|rrr & a & b & c

1 & 1 & 1 & 1

2 & 1 & -1 & -1

2 & 2 & 1 & 0

r_ij = -

r_(i-2)0 & r_(i-2)(j+1)

r_(i-1)0 & r_(i-1)(j+1)

= .