Dynamic Equations for the 3-DOF Helicopter

© 2004 Quanser Consulting Inc.

URL: http://www.quanser.com

Description

The Lagrange's method is used to obtain the dynamic model of the system.

The Quanser_Tools Package

The Quanser_Tools module defines generic procedures and data in relation to determining the state-space representation of all the Quanser experiments. Specifically, this means deriving and solving the Lagrange's equations of the Quanser systems.

The quanser repository containing the Quanser_Tools package is implemented in the 2 following files: quanser.ind and quanser.lib. If these two files are not readily available, they can be generated by executing the Maple worksheet titled: quanser_tools.mws.

To install the Quanser_Tools package, copy the two files quanser.ind and quanser.lib  into a directory of your choice, like for example: "C:\Program Files\Quanser\Maple Repository".

To use the Quanser_Tools package in a Maple worksheet, add the path to its disk location to the Maple global variable libname. For example, this can be achieved by the following Maple command:
libname :=
"C:/Program Files/Quanser/Maple Repository", libname:

Worksheet Initialization

> restart: interface( imaginaryunit = j ):

> with( LinearAlgebra ):

> libname := "C:/Program Files/Quanser/Maple Repository", ".", libname:

> with( Quanser_Tools );

[HTM, deriveA, deriveB, deriveF, kinetic_energy, lagrange_equations, moment_of_inertia, n_norm, potential_energy, write_ABCD_to_Mfile]

environment variable representing the order of series calculations

> Order := 2:

Notations

Generalized Coordinates: q[i] 's

The generalized coordinates are also called Lagrangian coordinates.

> q := [ epsilon(t), p(t), lambda(t) ];

q := [epsilon(t), p(t), lambda(t)]

Nq = number of Lagrangian coordinates

Nq is also the number of position states.

> Nq := nops( q ):

qd = first-order time derivative of the generalized coordinates

> qd := map( diff, q, t );

qd := [diff(epsilon(t), t), diff(p(t), t), diff(lambda(t), t)]

Cartesian Coordinates of the Moving Bodies

> HTM_BASE_TO_TRAVEL := HTM( 'rot', 'Z', -lambda(t) );

HTM_BASE_TO_TRAVEL := Matrix([[cos(lambda(t)), sin(lambda(t)), 0, 0], [-sin(lambda(t)), cos(lambda(t)), 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]])

> HTM_TRAVEL_TO_CW := Multiply( HTM( 'rot', 'X', epsilon(t) ), HTM( 'trans', 0, -L[w], 0 ) );

HTM_TRAVEL_TO_CW := Matrix([[1, 0, 0, 0], [0, cos(epsilon(t)), -sin(epsilon(t)), -cos(epsilon(t))*L[w]], [0, sin(epsilon(t)), cos(epsilon(t)), -sin(epsilon(t))*L[w]], [0, 0, 0, 1]])

> HTM_TRAVEL_TO_HB := Multiply( HTM( 'rot', 'X', epsilon(t) ), HTM( 'trans', 0, L[a], 0 ) );

HTM_TRAVEL_TO_HB := Matrix([[1, 0, 0, 0], [0, cos(epsilon(t)), -sin(epsilon(t)), cos(epsilon(t))*L[a]], [0, sin(epsilon(t)), cos(epsilon(t)), sin(epsilon(t))*L[a]], [0, 0, 0, 1]])

> HTM_HB_TO_FM := Multiply( HTM( 'rot', 'Y', -p(t) ), HTM( 'trans', L[h], 0, 0 ) );

HTM_HB_TO_FM := Matrix([[cos(p(t)), 0, -sin(p(t)), cos(p(t))*L[h]], [0, 1, 0, 0], [sin(p(t)), 0, cos(p(t)), sin(p(t))*L[h]], [0, 0, 0, 1]])

> HTM_HB_TO_BM := Multiply( HTM( 'rot', 'Y', -p(t) ), HTM( 'trans', -L[h], 0, 0 ) );

HTM_HB_TO_BM := Matrix([[cos(p(t)), 0, -sin(p(t)), -cos(p(t))*L[h]], [0, 1, 0, 0], [sin(p(t)), 0, cos(p(t)), -sin(p(t))*L[h]], [0, 0, 0, 1]])

> HTM_BASE_TO_CW := Multiply( HTM_BASE_TO_TRAVEL, HTM_TRAVEL_TO_CW );

HTM_BASE_TO_CW := Matrix([[cos(lambda(t)), sin(lambda(t))*cos(epsilon(t)), -sin(lambda(t))*sin(epsilon(t)), -sin(lambda(t))*cos(epsilon(t))*L[w]], [-sin(lambda(t)), cos(lambda(t))*cos(epsilon(t)), -co...

> HTM_BASE_TO_HB := Multiply( HTM_BASE_TO_TRAVEL, HTM_TRAVEL_TO_HB );

HTM_BASE_TO_HB := Matrix([[cos(lambda(t)), sin(lambda(t))*cos(epsilon(t)), -sin(lambda(t))*sin(epsilon(t)), sin(lambda(t))*cos(epsilon(t))*L[a]], [-sin(lambda(t)), cos(lambda(t))*cos(epsilon(t)), -cos...

> HTM_BASE_TO_FM := Multiply( HTM_BASE_TO_HB, HTM_HB_TO_FM );

HTM_BASE_TO_FM := Matrix([[cos(lambda(t))*cos(p(t))-sin(lambda(t))*sin(epsilon(t))*sin(p(t)), sin(lambda(t))*cos(epsilon(t)), -cos(lambda(t))*sin(p(t))-sin(lambda(t))*sin(epsilon(t))*cos(p(t)), cos(la...HTM_BASE_TO_FM := Matrix([[cos(lambda(t))*cos(p(t))-sin(lambda(t))*sin(epsilon(t))*sin(p(t)), sin(lambda(t))*cos(epsilon(t)), -cos(lambda(t))*sin(p(t))-sin(lambda(t))*sin(epsilon(t))*cos(p(t)), cos(la...HTM_BASE_TO_FM := Matrix([[cos(lambda(t))*cos(p(t))-sin(lambda(t))*sin(epsilon(t))*sin(p(t)), sin(lambda(t))*cos(epsilon(t)), -cos(lambda(t))*sin(p(t))-sin(lambda(t))*sin(epsilon(t))*cos(p(t)), cos(la...HTM_BASE_TO_FM := Matrix([[cos(lambda(t))*cos(p(t))-sin(lambda(t))*sin(epsilon(t))*sin(p(t)), sin(lambda(t))*cos(epsilon(t)), -cos(lambda(t))*sin(p(t))-sin(lambda(t))*sin(epsilon(t))*cos(p(t)), cos(la...

> HTM_BASE_TO_BM := Multiply( HTM_BASE_TO_HB, HTM_HB_TO_BM );

HTM_BASE_TO_BM := Matrix([[cos(lambda(t))*cos(p(t))-sin(lambda(t))*sin(epsilon(t))*sin(p(t)), sin(lambda(t))*cos(epsilon(t)), -cos(lambda(t))*sin(p(t))-sin(lambda(t))*sin(epsilon(t))*cos(p(t)), -cos(l...HTM_BASE_TO_BM := Matrix([[cos(lambda(t))*cos(p(t))-sin(lambda(t))*sin(epsilon(t))*sin(p(t)), sin(lambda(t))*cos(epsilon(t)), -cos(lambda(t))*sin(p(t))-sin(lambda(t))*sin(epsilon(t))*cos(p(t)), -cos(l...HTM_BASE_TO_BM := Matrix([[cos(lambda(t))*cos(p(t))-sin(lambda(t))*sin(epsilon(t))*sin(p(t)), sin(lambda(t))*cos(epsilon(t)), -cos(lambda(t))*sin(p(t))-sin(lambda(t))*sin(epsilon(t))*cos(p(t)), -cos(l...HTM_BASE_TO_BM := Matrix([[cos(lambda(t))*cos(p(t))-sin(lambda(t))*sin(epsilon(t))*sin(p(t)), sin(lambda(t))*cos(epsilon(t)), -cos(lambda(t))*sin(p(t))-sin(lambda(t))*sin(epsilon(t))*cos(p(t)), -cos(l...

> x[cw] := HTM_BASE_TO_CW[ 1, 4 ];
y[cw] := HTM_BASE_TO_CW[ 2, 4 ];

z[cw] := HTM_BASE_TO_CW[ 3, 4 ];

x[cw] := -sin(lambda(t))*cos(epsilon(t))*L[w]

y[cw] := -cos(lambda(t))*cos(epsilon(t))*L[w]

z[cw] := -sin(epsilon(t))*L[w]

> x[fm] := HTM_BASE_TO_FM[ 1, 4 ];
y[fm] := HTM_BASE_TO_FM[ 2, 4 ];

z[fm] := HTM_BASE_TO_FM[ 3, 4 ];

x[fm] := cos(lambda(t))*cos(p(t))*L[h]-sin(lambda(t))*sin(epsilon(t))*sin(p(t))*L[h]+sin(lambda(t))*cos(epsilon(t))*L[a]

y[fm] := -sin(lambda(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(epsilon(t))*sin(p(t))*L[h]+cos(lambda(t))*cos(epsilon(t))*L[a]

z[fm] := cos(epsilon(t))*sin(p(t))*L[h]+sin(epsilon(t))*L[a]

> x[bm] := HTM_BASE_TO_BM[ 1, 4 ];
y[bm] := HTM_BASE_TO_BM[ 2, 4 ];

z[bm] := HTM_BASE_TO_BM[ 3, 4 ];

x[bm] := -cos(lambda(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(epsilon(t))*sin(p(t))*L[h]+sin(lambda(t))*cos(epsilon(t))*L[a]

y[bm] := sin(lambda(t))*cos(p(t))*L[h]+cos(lambda(t))*sin(epsilon(t))*sin(p(t))*L[h]+cos(lambda(t))*cos(epsilon(t))*L[a]

z[bm] := -cos(epsilon(t))*sin(p(t))*L[h]+sin(epsilon(t))*L[a]

> xd[cw] := diff( x[cw], t );
yd[cw] := diff( y[cw], t );

zd[cw] := diff( z[cw], t );

xd[cw] := -cos(lambda(t))*diff(lambda(t), t)*cos(epsilon(t))*L[w]+sin(lambda(t))*sin(epsilon(t))*diff(epsilon(t), t)*L[w]

yd[cw] := sin(lambda(t))*diff(lambda(t), t)*cos(epsilon(t))*L[w]+cos(lambda(t))*sin(epsilon(t))*diff(epsilon(t), t)*L[w]

zd[cw] := -cos(epsilon(t))*diff(epsilon(t), t)*L[w]

> xd[fm] := diff( x[fm], t );
yd[fm] := diff( y[fm], t );

zd[fm] := diff( z[fm], t );

xd[fm] := -sin(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]-cos(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]-sin(lambda(t))*cos(epsilon(t))*...xd[fm] := -sin(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]-cos(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]-sin(lambda(t))*cos(epsilon(t))*...

yd[fm] := -cos(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]+sin(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]-cos(lambda(t))*cos(epsilon(t))*...yd[fm] := -cos(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]+sin(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]-cos(lambda(t))*cos(epsilon(t))*...

zd[fm] := -sin(epsilon(t))*diff(epsilon(t), t)*sin(p(t))*L[h]+cos(epsilon(t))*cos(p(t))*diff(p(t), t)*L[h]+cos(epsilon(t))*diff(epsilon(t), t)*L[a]

> xd[bm] := diff( x[bm], t );
yd[bm] := diff( y[bm], t );

zd[bm] := diff( z[bm], t );

xd[bm] := sin(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]+cos(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]+cos(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]+sin(lambda(t))*cos(epsilon(t))*d...xd[bm] := sin(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]+cos(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]+cos(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]+sin(lambda(t))*cos(epsilon(t))*d...

yd[bm] := cos(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]-sin(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]-sin(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]+cos(lambda(t))*cos(epsilon(t))*d...yd[bm] := cos(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]-sin(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]-sin(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]+cos(lambda(t))*cos(epsilon(t))*d...

zd[bm] := sin(epsilon(t))*diff(epsilon(t), t)*sin(p(t))*L[h]-cos(epsilon(t))*cos(p(t))*diff(p(t), t)*L[h]+cos(epsilon(t))*diff(epsilon(t), t)*L[a]

State-Space Variables

The chosen states should at least include the generalized coordinates and their first-time derivatives.

X is the state vector.

In the state vector X: Lagrangian coordinates are first, followed by their first-time derivatives, and finally any other states, as required.

Substitution sets for the states (to obtain time-independent state equations).

> subs_Xq := { seq( q[i] = X[i], i=1..Nq ) };
subs_Xqd := { seq( qd[i] = X[i+Nq], i=1..Nq ) };

subs_Xq := {epsilon(t) = X[1], p(t) = X[2], lambda(t) = X[3]}

subs_Xqd := {diff(lambda(t), t) = X[6], diff(epsilon(t), t) = X[4], diff(p(t), t) = X[5]}

Substitution set for the input(s).

> subs_U := { V[f] = U[1], V[b] = U[2] };

subs_U := {V[f] = U[1], V[b] = U[2]}

Nu = number of inputs; U = input (row) vector

> Nu := nops( subs_U ):

substitution set for the position states' second time derivatives

> subs_Xqdd := { seq( diff( q[i], t$2 ) =  Xd[i+Nq], i=1..Nq ) };

subs_Xqdd := {diff(lambda(t), `$`(t, 2)) = Xd[6], diff(p(t), `$`(t, 2)) = Xd[5], diff(epsilon(t), `$`(t, 2)) = Xd[4]}

second time derivatives of the position states (written as time-independent variables).

The set of unknowns is obtained from this list to solve the Lagrange's equations of motion.

> Xqdd := [ seq( Xd[i+Nq], i=1..Nq ) ];

Xqdd := [Xd[4], Xd[5], Xd[6]]

substitution set to linearize the state-space matrices (i.e. A and B)

about the quiescent null state vector (small-displacement theory)

> subs_XU_op := { seq( X[i] = 0, i=1..2*Nq ), seq( U[i] = 0, i=1..Nu ) }:

Nx = dim( X ) = total number of states (should be greater than or equal to: 2 * Nq)

Ny = chosen number of outputs

> Nx := 2 * Nq + 0:
Ny := Nq:

Total Potential and Kinetic Energies of the System

The total potential and kinetic energies are needed to calculate the Lagrangian of the system.

Total Potential Energy: V[T]

The total potential energy can be expressed in terms of the generalized coordinates alone.

Ve[T] = Total Elastic Potential Energy of the system

> Ve[T] := 0:

Vg[T] = Total Gravitational Potential Energy of the system

initialization:

> Vg[cw] := potential_energy( 'gravity', M[c], g, z[cw] );

Vg[cw] := -M[c]*g*sin(epsilon(t))*L[w]

> Vg[fm] := potential_energy( 'gravity', M[f], g, z[fm] );

Vg[fm] := M[f]*g*(cos(epsilon(t))*sin(p(t))*L[h]+sin(epsilon(t))*L[a])

> Vg[bm] := potential_energy( 'gravity', M[b], g, z[bm] );

Vg[bm] := M[b]*g*(-cos(epsilon(t))*sin(p(t))*L[h]+sin(epsilon(t))*L[a])

> Vg[T] := Vg[cw] + Vg[fm] + Vg[bm]:

V[T] = Total Potential Energy of the system

> V[T] := simplify( Ve[T] + Vg[T] );

V[T] := -g*(M[c]*sin(epsilon(t))*L[w]-M[f]*cos(epsilon(t))*sin(p(t))*L[h]-M[f]*sin(epsilon(t))*L[a]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]-M[b]*sin(epsilon(t))*L[a])

Total Kinetic Energy: T[T]

The total kinetic energy can be expressed in terms of the generalized coordinates and their first-time derivatives.

> v[cw] := n_norm( [ xd[cw], yd[cw], zd[cw] ], 2 ):

> Tt[cw] := kinetic_energy( 'translation', M[c], v[cw] );

Tt[cw] := 1/2*M[c]*((-cos(lambda(t))*diff(lambda(t), t)*cos(epsilon(t))*L[w]+sin(lambda(t))*sin(epsilon(t))*diff(epsilon(t), t)*L[w])^2+(sin(lambda(t))*diff(lambda(t), t)*cos(epsilon(t))*L[w]+cos(lamb...Tt[cw] := 1/2*M[c]*((-cos(lambda(t))*diff(lambda(t), t)*cos(epsilon(t))*L[w]+sin(lambda(t))*sin(epsilon(t))*diff(epsilon(t), t)*L[w])^2+(sin(lambda(t))*diff(lambda(t), t)*cos(epsilon(t))*L[w]+cos(lamb...

> v[fm] := n_norm( [ xd[fm], yd[fm], zd[fm] ], 2 ):

> Tt[fm] := kinetic_energy( 'translation', M[f], v[fm] );

Tt[fm] := 1/2*M[f]*((-sin(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]-cos(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]-sin(lambda(t))*cos(e...Tt[fm] := 1/2*M[f]*((-sin(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]-cos(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]-sin(lambda(t))*cos(e...Tt[fm] := 1/2*M[f]*((-sin(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]-cos(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]-sin(lambda(t))*cos(e...Tt[fm] := 1/2*M[f]*((-sin(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]-cos(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]-sin(lambda(t))*cos(e...Tt[fm] := 1/2*M[f]*((-sin(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]-cos(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]-sin(lambda(t))*cos(e...

> v[bm] := n_norm( [ xd[bm], yd[bm], zd[bm] ], 2 ):

> Tt[bm] := kinetic_energy( 'translation', M[b], v[bm] );

Tt[bm] := 1/2*M[b]*((sin(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]+cos(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]+cos(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]+sin(lambda(t))*cos(ep...Tt[bm] := 1/2*M[b]*((sin(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]+cos(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]+cos(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]+sin(lambda(t))*cos(ep...Tt[bm] := 1/2*M[b]*((sin(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]+cos(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]+cos(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]+sin(lambda(t))*cos(ep...Tt[bm] := 1/2*M[b]*((sin(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]+cos(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]+cos(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]+sin(lambda(t))*cos(ep...Tt[bm] := 1/2*M[b]*((sin(lambda(t))*diff(lambda(t), t)*cos(p(t))*L[h]+cos(lambda(t))*sin(p(t))*diff(p(t), t)*L[h]+cos(lambda(t))*diff(lambda(t), t)*sin(epsilon(t))*sin(p(t))*L[h]+sin(lambda(t))*cos(ep...

T[T] = Total Kinetic Energy of the system

> T[T] := Tt[cw] + Tt[fm] + Tt[bm]:
T[T] := collect( T[T], diff );

T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...T[T] := (1/2*M[f]*((-cos(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]+sin(lambda(t))*sin(p(t))*L[h])^2+(-sin(lambda(t))*sin(epsilon(t))*cos(p(t))*L[h]-cos(lambda(t))*sin(p(t))*L[h])^2+cos(epsilon(t))^2*c...

Generalized Forces: Q[i] 's

Qe: on epsilon: elevation

> Q[1] := L[a] * K[m] * ( V[f] + V[b] );

Q[1] := L[a]*K[m]*(V[f]+V[b])

Qp: on pitch, p

> Q[2] := K[m] * ( V[f] - V[b] ) * L[h];

Q[2] := K[m]*(V[f]-V[b])*L[h]

Qt: on travel, lambda

> Q[3] := L[a] * K[m] * ( V[f] + V[b] ) * sin( p(t) );

Q[3] := L[a]*K[m]*(V[f]+V[b])*sin(p(t))

> Q := [ seq( Q[i], i=1..Nq ) ];

Q := [L[a]*K[m]*(V[f]+V[b]), K[m]*(V[f]-V[b])*L[h], L[a]*K[m]*(V[f]+V[b])*sin(p(t))]

quiescent voltage: operating point

> EOM_OP_PT := L[w] * M[c] * g = L[a] * g * ( M[f] + M[b] ) + L[a] * ( V[op] + V[op] ) * K[m];

EOM_OP_PT := L[w]*M[c]*g = L[a]*g*(M[f]+M[b])+2*L[a]*V[op]*K[m]

> V[op] := solve( EOM_OP_PT, V[op] );

V[op] := 1/2*g*(L[w]*M[c]-L[a]*M[f]-L[a]*M[b])/(L[a]*K[m])

> subs_U_op := { V[f] = V[f] - V[op], V[b] = V[b] - V[op] }:

> Q := subs( subs_U_op, Q );

Q := [L[a]*K[m]*(V[f]-g*(L[w]*M[c]-L[a]*M[f]-L[a]*M[b])/(L[a]*K[m])+V[b]), K[m]*(V[f]-V[b])*L[h], L[a]*K[m]*(V[f]-g*(L[w]*M[c]-L[a]*M[f]-L[a]*M[b])/(L[a]*K[m])+V[b])*sin(p(t))]

Euler-Lagrange's Equations

For a N-DOF system, the Lagrange's equations can be written:

Diff(Diff(L, qdot[i]), t)-Diff(L, q[i]) = Q[i]  for i = 1 through N

where:

Q[i] 's are special combinations of external forces and called the generalized forces,

q[1] , ..., q[N] , are N independent coordinates chosen to describe the system and called the generalized coordinates,

and L is the Lagrangian of the system.

L is defined by:

L = T-U

where T is the total kinetic energy of the system and U the total potential energy of the system.

> EOM_orig := lagrange_equations( T[T], V[T], q, Q ):

this is to display the EOM's

> EOM_orig := collect( EOM_orig, { seq( diff( q[i], t$2 ), i=1..Nq ), seq( diff( q[i], t ), i=1..Nq ), seq( q[i], i=1..Nq ) } );

EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...EOM_orig := [(-M[b]*L[a]*cos(p(t))*L[h]+M[f]*L[a]*cos(p(t))*L[h])*diff(p(t), `$`(t, 2))+(-M[b]*sin(epsilon(t))*L[a]*cos(p(t))*L[h]+M[b]*cos(epsilon(t))*sin(p(t))*L[h]^2*cos(p(t))+M[f]*cos(epsilon(t))*...

Express the Euler-Lagrange equations of motion as functions of the states:

1) substitute (i.e. name) the acceleration states first!

2) then substitute the velocity states!

3) and only after, the position states, and the inputs!

> EOM_states := subs( subs_Xqd, subs( subs_Xqdd, EOM_orig ) ):

> EOM_states := subs( subs_Xq, subs_U, EOM_states );

EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...EOM_states := [(-M[b]*L[a]*cos(X[2])*L[h]+M[f]*L[a]*cos(X[2])*L[h])*Xd[5]+(-M[b]*sin(X[1])*L[a]*cos(X[2])*L[h]+M[b]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*cos(X[1])*sin(X[2])*L[h]^2*cos(X[2])+M[f]*...

Linearization  in the EOM's of the Trigonometric Functions

Linearization of the equations of motion around the quiescent point of operation (in order to solve them).

Here, linearization around the zero angles, i.e. for small-amplitude oscillations.

Linearization around: alpha0 = 0, and alpha_dot0 = 0

> EOM_ser := EOM_states:

Generalized series expansions of the trigonometric functions is used (for small angles).

> for i from 1 to Nq do
 EOM_ser[i] := subsop( 1 = convert( series( op( 1, EOM_ser[i] ), X[1] ), polynom ), EOM_ser[i] );

 EOM_ser[i] := subsop( 1 = convert( series( op( 1, EOM_ser[i] ), X[2] ), polynom ), EOM_ser[i] );

 EOM_ser[i] := subsop( 1 = convert( series( op( 1, EOM_ser[i] ), X[3] ), polynom ), EOM_ser[i] );

 EOM_ser[i] := simplify( EOM_ser[i] );

end do:

> EOM_ser;

[L[a]*g*M[f]+Xd[4]*M[b]*L[a]^2+Xd[4]*M[c]*L[w]^2+Xd[4]*M[f]*L[a]^2-L[w]*M[c]*g-Xd[5]*L[a]*M[b]*L[h]+Xd[5]*L[a]*M[f]*L[h]+X[1]*X[6]^2*M[c]*L[w]^2+X[1]*X[6]^2*M[f]*L[a]^2+X[1]*X[6]^2*M[b]*L[a]^2+X[2]*Xd...[L[a]*g*M[f]+Xd[4]*M[b]*L[a]^2+Xd[4]*M[c]*L[w]^2+Xd[4]*M[f]*L[a]^2-L[w]*M[c]*g-Xd[5]*L[a]*M[b]*L[h]+Xd[5]*L[a]*M[f]*L[h]+X[1]*X[6]^2*M[c]*L[w]^2+X[1]*X[6]^2*M[f]*L[a]^2+X[1]*X[6]^2*M[b]*L[a]^2+X[2]*Xd...[L[a]*g*M[f]+Xd[4]*M[b]*L[a]^2+Xd[4]*M[c]*L[w]^2+Xd[4]*M[f]*L[a]^2-L[w]*M[c]*g-Xd[5]*L[a]*M[b]*L[h]+Xd[5]*L[a]*M[f]*L[h]+X[1]*X[6]^2*M[c]*L[w]^2+X[1]*X[6]^2*M[f]*L[a]^2+X[1]*X[6]^2*M[b]*L[a]^2+X[2]*Xd...[L[a]*g*M[f]+Xd[4]*M[b]*L[a]^2+Xd[4]*M[c]*L[w]^2+Xd[4]*M[f]*L[a]^2-L[w]*M[c]*g-Xd[5]*L[a]*M[b]*L[h]+Xd[5]*L[a]*M[f]*L[h]+X[1]*X[6]^2*M[c]*L[w]^2+X[1]*X[6]^2*M[f]*L[a]^2+X[1]*X[6]^2*M[b]*L[a]^2+X[2]*Xd...[L[a]*g*M[f]+Xd[4]*M[b]*L[a]^2+Xd[4]*M[c]*L[w]^2+Xd[4]*M[f]*L[a]^2-L[w]*M[c]*g-Xd[5]*L[a]*M[b]*L[h]+Xd[5]*L[a]*M[f]*L[h]+X[1]*X[6]^2*M[c]*L[w]^2+X[1]*X[6]^2*M[f]*L[a]^2+X[1]*X[6]^2*M[b]*L[a]^2+X[2]*Xd...[L[a]*g*M[f]+Xd[4]*M[b]*L[a]^2+Xd[4]*M[c]*L[w]^2+Xd[4]*M[f]*L[a]^2-L[w]*M[c]*g-Xd[5]*L[a]*M[b]*L[h]+Xd[5]*L[a]*M[f]*L[h]+X[1]*X[6]^2*M[c]*L[w]^2+X[1]*X[6]^2*M[f]*L[a]^2+X[1]*X[6]^2*M[b]*L[a]^2+X[2]*Xd...[L[a]*g*M[f]+Xd[4]*M[b]*L[a]^2+Xd[4]*M[c]*L[w]^2+Xd[4]*M[f]*L[a]^2-L[w]*M[c]*g-Xd[5]*L[a]*M[b]*L[h]+Xd[5]*L[a]*M[f]*L[h]+X[1]*X[6]^2*M[c]*L[w]^2+X[1]*X[6]^2*M[f]*L[a]^2+X[1]*X[6]^2*M[b]*L[a]^2+X[2]*Xd...[L[a]*g*M[f]+Xd[4]*M[b]*L[a]^2+Xd[4]*M[c]*L[w]^2+Xd[4]*M[f]*L[a]^2-L[w]*M[c]*g-Xd[5]*L[a]*M[b]*L[h]+Xd[5]*L[a]*M[f]*L[h]+X[1]*X[6]^2*M[c]*L[w]^2+X[1]*X[6]^2*M[f]*L[a]^2+X[1]*X[6]^2*M[b]*L[a]^2+X[2]*Xd...[L[a]*g*M[f]+Xd[4]*M[b]*L[a]^2+Xd[4]*M[c]*L[w]^2+Xd[4]*M[f]*L[a]^2-L[w]*M[c]*g-Xd[5]*L[a]*M[b]*L[h]+Xd[5]*L[a]*M[f]*L[h]+X[1]*X[6]^2*M[c]*L[w]^2+X[1]*X[6]^2*M[f]*L[a]^2+X[1]*X[6]^2*M[b]*L[a]^2+X[2]*Xd...[L[a]*g*M[f]+Xd[4]*M[b]*L[a]^2+Xd[4]*M[c]*L[w]^2+Xd[4]*M[f]*L[a]^2-L[w]*M[c]*g-Xd[5]*L[a]*M[b]*L[h]+Xd[5]*L[a]*M[f]*L[h]+X[1]*X[6]^2*M[c]*L[w]^2+X[1]*X[6]^2*M[f]*L[a]^2+X[1]*X[6]^2*M[b]*L[a]^2+X[2]*Xd...

Additional Insight: Inertia (or mass) Matrix: Fi

The nonlinear system of equations resulting from the Lagrangian mechanics can be written in the following matrix form:

F( q ) . qdd + G( q, qd ) . qd + H( q ) . q = L( q, qd, u )

F, G, and H are called, respectively, the mass, damping, and stiffness matrices.

They are symmetric in form.

The inertia (a.k.a. mass) matrix, F, gives indications regarding the coupling existing in the system.

> Fi := Matrix( Nq, Nq ):

> for i from 1 to Nq do
 for k from 1 to Nq do

   Fi[ i, k ] := simplify( diff( op( 1, EOM_states[i] ), Xd[k+Nq] ) );

   Fi[ i, k ] := collect( combine( Fi[ i, k ], trig ), cos );

 end do;

end do:

> 'F[i]' = Fi;

F[i] = Matrix([[(-1/2*M[f]*L[h]^2-1/2*M[b]*L[h]^2)*cos(2*X[2])+M[f]*L[a]^2+1/2*M[f]*L[h]^2+1/2*M[b]*L[h]^2+M[c]*L[w]^2+M[b]*L[a]^2, (-L[a]*M[b]*L[h]+L[a]*M[f]*L[h])*cos(X[2]), -1/2*L[a]*M[b]*L[h]*sin(...F[i] = Matrix([[(-1/2*M[f]*L[h]^2-1/2*M[b]*L[h]^2)*cos(2*X[2])+M[f]*L[a]^2+1/2*M[f]*L[h]^2+1/2*M[b]*L[h]^2+M[c]*L[w]^2+M[b]*L[a]^2, (-L[a]*M[b]*L[h]+L[a]*M[f]*L[h])*cos(X[2]), -1/2*L[a]*M[b]*L[h]*sin(...F[i] = Matrix([[(-1/2*M[f]*L[h]^2-1/2*M[b]*L[h]^2)*cos(2*X[2])+M[f]*L[a]^2+1/2*M[f]*L[h]^2+1/2*M[b]*L[h]^2+M[c]*L[w]^2+M[b]*L[a]^2, (-L[a]*M[b]*L[h]+L[a]*M[f]*L[h])*cos(X[2]), -1/2*L[a]*M[b]*L[h]*sin(...F[i] = Matrix([[(-1/2*M[f]*L[h]^2-1/2*M[b]*L[h]^2)*cos(2*X[2])+M[f]*L[a]^2+1/2*M[f]*L[h]^2+1/2*M[b]*L[h]^2+M[c]*L[w]^2+M[b]*L[a]^2, (-L[a]*M[b]*L[h]+L[a]*M[f]*L[h])*cos(X[2]), -1/2*L[a]*M[b]*L[h]*sin(...F[i] = Matrix([[(-1/2*M[f]*L[h]^2-1/2*M[b]*L[h]^2)*cos(2*X[2])+M[f]*L[a]^2+1/2*M[f]*L[h]^2+1/2*M[b]*L[h]^2+M[c]*L[w]^2+M[b]*L[a]^2, (-L[a]*M[b]*L[h]+L[a]*M[f]*L[h])*cos(X[2]), -1/2*L[a]*M[b]*L[h]*sin(...F[i] = Matrix([[(-1/2*M[f]*L[h]^2-1/2*M[b]*L[h]^2)*cos(2*X[2])+M[f]*L[a]^2+1/2*M[f]*L[h]^2+1/2*M[b]*L[h]^2+M[c]*L[w]^2+M[b]*L[a]^2, (-L[a]*M[b]*L[h]+L[a]*M[f]*L[h])*cos(X[2]), -1/2*L[a]*M[b]*L[h]*sin(...F[i] = Matrix([[(-1/2*M[f]*L[h]^2-1/2*M[b]*L[h]^2)*cos(2*X[2])+M[f]*L[a]^2+1/2*M[f]*L[h]^2+1/2*M[b]*L[h]^2+M[c]*L[w]^2+M[b]*L[a]^2, (-L[a]*M[b]*L[h]+L[a]*M[f]*L[h])*cos(X[2]), -1/2*L[a]*M[b]*L[h]*sin(...F[i] = Matrix([[(-1/2*M[f]*L[h]^2-1/2*M[b]*L[h]^2)*cos(2*X[2])+M[f]*L[a]^2+1/2*M[f]*L[h]^2+1/2*M[b]*L[h]^2+M[c]*L[w]^2+M[b]*L[a]^2, (-L[a]*M[b]*L[h]+L[a]*M[f]*L[h])*cos(X[2]), -1/2*L[a]*M[b]*L[h]*sin(...F[i] = Matrix([[(-1/2*M[f]*L[h]^2-1/2*M[b]*L[h]^2)*cos(2*X[2])+M[f]*L[a]^2+1/2*M[f]*L[h]^2+1/2*M[b]*L[h]^2+M[c]*L[w]^2+M[b]*L[a]^2, (-L[a]*M[b]*L[h]+L[a]*M[f]*L[h])*cos(X[2]), -1/2*L[a]*M[b]*L[h]*sin(...F[i] = Matrix([[(-1/2*M[f]*L[h]^2-1/2*M[b]*L[h]^2)*cos(2*X[2])+M[f]*L[a]^2+1/2*M[f]*L[h]^2+1/2*M[b]*L[h]^2+M[c]*L[w]^2+M[b]*L[a]^2, (-L[a]*M[b]*L[h]+L[a]*M[f]*L[h])*cos(X[2]), -1/2*L[a]*M[b]*L[h]*sin(...

Linearization of the inertia matrix for small-displacements

> Fi_lin := Matrix( Nq, Nq ):

> for i from 1 to Nq do
 for k from 1 to Nq do

   Fi_lin[ i, k ] := Fi[ i, k ];

   Fi_lin[ i, k ] := convert( series( Fi_lin[ i, k ], X[ 2 ] ), polynom );

   Fi_lin[ i, k ] := subs( subs_XU_op, Fi_lin[ i, k ] );

 end do;

end do:

> 'F_lin' = Fi_lin;

F_lin = Matrix([[M[f]*L[a]^2+M[c]*L[w]^2+M[b]*L[a]^2, -L[a]*M[b]*L[h]+L[a]*M[f]*L[h], 0], [-L[a]*M[b]*L[h]+L[a]*M[f]*L[h], M[f]*L[h]^2+M[b]*L[h]^2, 0], [0, 0, M[f]*L[a]^2+M[b]*L[h]^2+M[c]*L[w]^2+M[b]*...

Solving the Euler-Lagrange's Equations

To solve the Euler-Lagrange's equations, they need to be linear.

Reverse State Substitution for Pretty Display of the Solved EOM's

only for pretty print

> subs_Xq_rev := { seq( X[i] = q[i], i=1..Nq ) }:
subs_Xqd_rev := { seq( X[i+Nq] = qd[i], i=1..Nq ) }:

> subs_U_rev := { U[1] = V[f], U[2] = V[b] };

subs_U_rev := {U[1] = V[f], U[2] = V[b]}

> eom_collect_list := { seq( diff( q[i], t ), i=1..Nq ), seq( q[i], i=1..Nq ) };

eom_collect_list := {epsilon(t), p(t), lambda(t), diff(p(t), t), diff(lambda(t), t), diff(epsilon(t), t)}

Solution to the Non-Linear Equations of Motion

Solve the non-linear form of the equations of motion for the states' second time derivatives

> Xqdd_solset_nl := solve( convert( EOM_states, set ), convert( Xqdd, set ) ):

> assign( Xqdd_solset_nl );

> Xd_nl[Nq+1] := simplify( Xd[Nq+1] ):
Xd_nl[Nq+2] := simplify( Xd[Nq+2] ):

Xd_nl[Nq+3] := simplify( Xd[Nq+3] ):

unassign( 'Xd[Nq+1]', 'Xd[Nq+2]', 'Xd[Nq+3]' ):

pretty display w.r.t. the named system states

> Xd_nl[Nq+1] := simplify( subs( subs_U_rev, subs_Xq_rev, subs_Xqd_rev, Xd_nl[Nq+1] ) ):
diff( qd[1], t ) = collect( Xd_nl[Nq+1], eom_collect_list );

diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...diff(epsilon(t), `$`(t, 2)) = -(-L[a]*M[f]^2*sin(p(t))*M[c]*L[w]^2*L[h]*cos(epsilon(t))-4*L[a]^3*M[f]^2*sin(p(t))*M[b]*L[h]*cos(epsilon(t))+4*L[a]^3*M[f]*M[b]^2*sin(p(t))*L[h]*cos(epsilon(t))+L[a]*M[b...

> Xd_nl[Nq+2] := simplify( subs( subs_U_rev, subs_Xq_rev, subs_Xqd_rev, Xd_nl[Nq+2] ) ):
diff( qd[2], t ) = collect( Xd_nl[Nq+2], eom_collect_list );

diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...diff(p(t), `$`(t, 2)) = -(-4*M[f]*cos(epsilon(t))*cos(p(t))*sin(epsilon(t))*L[a]^3*M[b]^2*L[h]^2-cos(epsilon(t))*cos(p(t))*L[w]^2*M[c]*sin(epsilon(t))*M[b]^2*L[a]*L[h]^2+M[f]^2*cos(epsilon(t))*cos(p(t...

> Xd_nl[Nq+3] := simplify( subs( subs_U_rev, subs_Xq_rev, subs_Xqd_rev, Xd_nl[Nq+3] ) ):
diff( qd[3], t ) = collect( Xd_nl[Nq+3], eom_collect_list );

diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...diff(lambda(t), `$`(t, 2)) = (-cos(p(t))*cos(epsilon(t))*M[c]*L[w]^2*L[a]*L[h]*M[b]^2+4*cos(p(t))*cos(epsilon(t))*M[f]^2*M[b]*L[a]^3*L[h]-4*cos(p(t))*cos(epsilon(t))*M[f]*M[b]^2*L[a]^3*L[h]+cos(p(t))*...

Solution to the Linearized EOM's

Solve the linear form of the equations of motion for the states' second time derivatives

> Xqdd_solset_ser := solve( convert( EOM_ser, set ), convert( Xqdd, set ) );

Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...Xqdd_solset_ser := {Xd[4] = -(-L[a]*M[f]^3*g*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[1]*L[h]^2-2*L[a]*M[f]*M[b]*K[m]*U[2]*L[h]^2-2*L[a]^3*M[f]*M[b]*K[m]*U[1]-M[b]^3*L[a]*g*L[h]^2-L[a]*M[f]^2*M[c]*L[w]^2*g-L[a]...

> assign( Xqdd_solset_ser );

Moreover, for small angles

> subs_small_angles_list := { X[1]^2 = 0, X[2]^2 = 0, X[3]^2 = 0, X[Nq+1]^2 = 0, X[Nq+2]^2 = 0, X[Nq+3]^2 = 0, M[b] = M[f] }:

> Xd[Nq+1] := algsubs( sin(X[2]) = X[2], Xd[Nq+1] ):

> Xd[Nq+1] := subs( subs_small_angles_list, Xd[Nq+1] ):

> Xd[Nq+1] := algsubs( X[Nq+1] * X[Nq+2] = 0, Xd[Nq+1] ):

> Xd[Nq+1] := algsubs( X[Nq+1] * X[Nq+3] = 0, Xd[Nq+1] ):

> Xd[Nq+1] := algsubs( X[Nq+2] * X[Nq+3] = 0, Xd[Nq+1] ):

> Xd[Nq+1] := algsubs( X[1] * X[2] = 0, Xd[Nq+1] ):

> Xd[Nq+1] := algsubs( X[1] * X[3] = 0, Xd[Nq+1] ):

> Xd[Nq+1] := algsubs( X[2] * X[3] = 0, Xd[Nq+1] );

Xd[4] := -(-4*L[a]^3*M[f]^2*U[2]*K[m]-2*L[a]*M[f]*M[c]*L[w]^2*K[m]*U[2]-4*L[a]^3*M[f]^2*K[m]*U[1]-2*L[a]*M[f]*M[c]*L[w]^2*K[m]*U[1]-4*L[a]*M[f]^2*K[m]*U[1]*L[h]^2-4*L[a]*M[f]^2*K[m]*U[2]*L[h]^2)/(8*M[...

> Xd[Nq+2] := algsubs( sin(X[2]) = X[2], Xd[Nq+2] ):

> Xd[Nq+2] := subs( subs_small_angles_list, Xd[Nq+2] ):

> Xd[Nq+2] := algsubs( X[Nq+1] * X[Nq+2] = 0, Xd[Nq+2] ):

> Xd[Nq+2] := algsubs( X[Nq+1] * X[Nq+3] = 0, Xd[Nq+2] ):

> Xd[Nq+2] := algsubs( X[Nq+2] * X[Nq+3] = 0, Xd[Nq+2] ):

> Xd[Nq+2] := algsubs( X[1] * X[2] = 0, Xd[Nq+2] ):

> Xd[Nq+2] := algsubs( X[1] * X[3] = 0, Xd[Nq+2] ):

> Xd[Nq+2] := algsubs( X[2] * X[3] = 0, Xd[Nq+2] );

Xd[5] := -(4*M[f]^2*K[m]*U[2]*L[a]^4-2*M[f]*K[m]*U[1]*M[c]*L[w]^2*L[h]^2+2*M[f]*K[m]*U[2]*M[c]*L[w]^2*L[h]^2+4*M[f]*K[m]*U[2]*M[c]*L[w]^2*L[a]^2-4*M[f]*K[m]*U[1]*M[c]*L[w]^2*L[a]^2-4*M[f]^2*U[1]*K[m]*...Xd[5] := -(4*M[f]^2*K[m]*U[2]*L[a]^4-2*M[f]*K[m]*U[1]*M[c]*L[w]^2*L[h]^2+2*M[f]*K[m]*U[2]*M[c]*L[w]^2*L[h]^2+4*M[f]*K[m]*U[2]*M[c]*L[w]^2*L[a]^2-4*M[f]*K[m]*U[1]*M[c]*L[w]^2*L[a]^2-4*M[f]^2*U[1]*K[m]*...

> Xd[Nq+3] := algsubs( sin(X[2]) = X[2], Xd[Nq+3] ):

> Xd[Nq+3] := subs( subs_small_angles_list, Xd[Nq+3] ):

> Xd[Nq+3] := algsubs( X[Nq+1] * X[Nq+2] = 0, Xd[Nq+3] ):

> Xd[Nq+3] := algsubs( X[Nq+1] * X[Nq+3] = 0, Xd[Nq+3] ):

> Xd[Nq+3] := algsubs( X[Nq+2] * X[Nq+3] = 0, Xd[Nq+3] ):

> Xd[Nq+3] := algsubs( X[1] * X[2] = 0, Xd[Nq+3] ):

> Xd[Nq+3] := algsubs( X[1] * X[3] = 0, Xd[Nq+3] ):

> Xd[Nq+3] := algsubs( X[2] * X[3] = 0, Xd[Nq+3] );

Xd[6] := (2*M[f]*(-2*M[f]*L[a]^2*L[h]*K[m]*U[1]+8*X[4]*M[f]^2*X[6]*L[a]^4+8*X[4]*M[f]*X[6]*M[c]*L[w]^2*L[a]^2+2*X[4]*X[6]*M[c]^2*L[w]^4-M[c]*L[w]^2*L[h]*K[m]*U[1]+M[c]*L[w]^2*L[h]*K[m]*U[2]+2*M[f]*L[a...Xd[6] := (2*M[f]*(-2*M[f]*L[a]^2*L[h]*K[m]*U[1]+8*X[4]*M[f]^2*X[6]*L[a]^4+8*X[4]*M[f]*X[6]*M[c]*L[w]^2*L[a]^2+2*X[4]*X[6]*M[c]^2*L[w]^4-M[c]*L[w]^2*L[h]*K[m]*U[1]+M[c]*L[w]^2*L[h]*K[m]*U[2]+2*M[f]*L[a...Xd[6] := (2*M[f]*(-2*M[f]*L[a]^2*L[h]*K[m]*U[1]+8*X[4]*M[f]^2*X[6]*L[a]^4+8*X[4]*M[f]*X[6]*M[c]*L[w]^2*L[a]^2+2*X[4]*X[6]*M[c]^2*L[w]^4-M[c]*L[w]^2*L[h]*K[m]*U[1]+M[c]*L[w]^2*L[h]*K[m]*U[2]+2*M[f]*L[a...

pretty display w.r.t. the named system states

1st Linearized Equation Of Motion in the time domain

> L_EOM_1_DT := diff( qd[1], t ) = collect( subs( subs_U_rev, subs_Xq_rev, subs_Xqd_rev, Xd[Nq+1] ), eom_collect_list ):
L_EOM_1_DT;

diff(epsilon(t), `$`(t, 2)) = -(-4*L[a]^3*M[f]^2*V[b]*K[m]-2*L[a]*M[f]*M[c]*L[w]^2*K[m]*V[b]-4*L[a]^3*M[f]^2*K[m]*V[f]-2*L[a]*M[f]*M[c]*L[w]^2*K[m]*V[f]-4*L[a]*M[f]^2*K[m]*V[f]*L[h]^2-4*L[a]*M[f]^2*K[...

2nd Linearized Equation Of Motion in the time domain

> L_EOM_2_DT := diff( qd[2], t ) = collect( subs( subs_U_rev, subs_Xq_rev, subs_Xqd_rev, Xd[Nq+2] ), eom_collect_list ):
L_EOM_2_DT;

diff(p(t), `$`(t, 2)) = -(4*M[f]^2*K[m]*V[b]*L[a]^4-2*M[f]*K[m]*V[f]*M[c]*L[w]^2*L[h]^2+2*M[f]*K[m]*V[b]*M[c]*L[w]^2*L[h]^2+4*M[f]*K[m]*V[b]*M[c]*L[w]^2*L[a]^2-4*M[f]*K[m]*V[f]*M[c]*L[w]^2*L[a]^2-4*M[...diff(p(t), `$`(t, 2)) = -(4*M[f]^2*K[m]*V[b]*L[a]^4-2*M[f]*K[m]*V[f]*M[c]*L[w]^2*L[h]^2+2*M[f]*K[m]*V[b]*M[c]*L[w]^2*L[h]^2+4*M[f]*K[m]*V[b]*M[c]*L[w]^2*L[a]^2-4*M[f]*K[m]*V[f]*M[c]*L[w]^2*L[a]^2-4*M[...

2nd Linearized Equation Of Motion (i.e., re. alpha) in the time domain

> L_EOM_3_DT := diff( qd[3], t ) = collect( subs( subs_U_rev, subs_Xq_rev, subs_Xqd_rev, Xd[Nq+3] ), eom_collect_list ):
L_EOM_3_DT;

diff(lambda(t), `$`(t, 2)) = (2*M[f]*(2*M[c]^2*L[w]^4+8*M[f]^2*L[a]^4+8*M[f]*M[c]*L[w]^2*L[a]^2)*diff(epsilon(t), t)*diff(lambda(t), t)/(8*M[f]^3*L[a]^2*L[h]^2+8*M[f]^3*L[a]^4+2*M[f]*M[c]^2*L[w]^4+4*M...diff(lambda(t), `$`(t, 2)) = (2*M[f]*(2*M[c]^2*L[w]^4+8*M[f]^2*L[a]^4+8*M[f]*M[c]*L[w]^2*L[a]^2)*diff(epsilon(t), t)*diff(lambda(t), t)/(8*M[f]^3*L[a]^2*L[h]^2+8*M[f]^3*L[a]^4+2*M[f]*M[c]^2*L[w]^4+4*M...diff(lambda(t), `$`(t, 2)) = (2*M[f]*(2*M[c]^2*L[w]^4+8*M[f]^2*L[a]^4+8*M[f]*M[c]*L[w]^2*L[a]^2)*diff(epsilon(t), t)*diff(lambda(t), t)/(8*M[f]^3*L[a]^2*L[h]^2+8*M[f]^3*L[a]^4+2*M[f]*M[c]^2*L[w]^4+4*M...diff(lambda(t), `$`(t, 2)) = (2*M[f]*(2*M[c]^2*L[w]^4+8*M[f]^2*L[a]^4+8*M[f]*M[c]*L[w]^2*L[a]^2)*diff(epsilon(t), t)*diff(lambda(t), t)/(8*M[f]^3*L[a]^2*L[h]^2+8*M[f]^3*L[a]^4+2*M[f]*M[c]^2*L[w]^4+4*M...

Determine the System State-Space Matrices: A, B, C, and D

> A_ss := Matrix( Nx, Nx ):

> A_ss := deriveA( Xqdd, A_ss, Nq, subs_XU_op ): 'A' = A_ss;

A = Matrix([[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, -(L[w]*M[c]-2*L[a]*M[f])*g/(M[c]*L[w]^2+2*M[f]*L[h]^2+2*M[f]*L[a]^2), 0, 0, 0, 0]])

> #lprint( A_ss );

> B_ss := Matrix( Nx, Nu ):

> B_ss := deriveB( Xqdd, B_ss, Nq, subs_XU_op ): 'B' = B_ss;

B = Matrix([[0, 0], [0, 0], [0, 0], [L[a]*K[m]/(M[c]*L[w]^2+2*M[f]*L[a]^2), L[a]*K[m]/(M[c]*L[w]^2+2*M[f]*L[a]^2)], [1/2*K[m]/(M[f]*L[h]), -1/2*K[m]/(M[f]*L[h])], [0, 0]])

> #lprint( B_ss );

> C_ss := IdentityMatrix( Nx, Nx );

C_ss := Matrix([[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]])

> #lprint( C_ss );

> D_ss := Matrix( Nx, Nu, 0 );

D_ss := Matrix([[0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0]])

> #lprint( D_ss );

Write A, B, C, and D to a Matlab file

Save the state-space matrices A, B, C and D to a MATLAB file.

> Matlab_File_Name := "HELI3D":

> Matlab_File_Name := cat( Matlab_File_Name, "_ABCD_eqns_VfVb.m" );

Matlab_File_Name :=

substitution set containing a notation consistent with that used in the MATLAB design script(s)

> Matlab_Notations := { L[w] = Lw, L[h] = Lh, M[f] = mf, L[a] = La, M[b] = mb, M[c] = mc, K[m] = Km, g = g }:

> Experiment_Name := "3-DOF Helicopter":

> write_ABCD_to_Mfile( Matlab_File_Name, Experiment_Name, Matlab_Notations, A_ss, B_ss, C_ss, D_ss );

Procedure Printing

default:

> #interface( verboseproc = 1 );

> #eval( lagrange_equations );

> #eval( deriveB );

>

>

>

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