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: 

Generalized Coordinates: 's 

The generalized coordinates are also called Lagrangian coordinates. 

=  the position of mass 1, the ground plane 

=  the position of mass 1 presenting a tire center of mass measured from is resting position 

position of mass 2, representing the car body measured from its resting position 

>	q1 := [ x[1](t), x[2](t) ,z[r](t) , x[1](t)-z[r](t), x[2](t)-x[1](t)];

 

(3.1.1)

 

>	q := [ x[1](t), x[2](t)  ];

 

(3.1.2)

 

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, e.g. position and angular velocities 

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

 

(3.1.3)

 

>	Nq := nops( q ):

 

>	u := z[r](t);

 

(3.1.4)

 

>	ud:= diff(u,t);

 

(3.1.5)

 

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 ) };

 

 

	(3.2.1)

 

Substitution set for the input(s). 

set the input to be the current cmd to the two motors and gravity 

>	subs_U := { ud = U[1],F[c]=U[2],g=U[3]};

 

(3.2.2)

 

Nu = number of inputs; U = input (row) vector (e.g. U = [ V[m] ] ) 

>	Nu := nops( subs_U );

 

(3.2.3)

 

substitution set for the position states' second time derivatives 

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

 

(3.2.4)

 

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 ) ];

 

(3.2.5)

 

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

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

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

 

(3.2.6)

 

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 Energy:  

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

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

>	Vg[1] := potential_energy( 'gravity', M[us], g, q[1]);

 

(4.1.1)

 

>	Vg[2] := potential_energy( 'gravity', M[s], g, q[2]);

 

(4.1.2)

 

>	Vg[T] := Vg[1] + Vg[2] ;

 

(4.1.3)

 

K[s] = spring torsional stiffness 

Ve[1] = first spring potetnial energy 

>	Ve[1] := potential_energy( 'spring', K[us], ( q[1] - u ) );

 

(4.1.4)

 

Ve[2] = second spring potetnial energy 

>	Ve[2] := potential_energy( 'spring', K[s], ( q[2] - q[1] ) );

 

(4.1.5)

 

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

>	Ve[T] := Ve[1] + Ve[2];

 

(4.1.6)

 

V[T] = Total Potential Energy of the system 

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

 

(4.1.7)

 

Total Kinetic Energy:  

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

= kinetic energy of the the Mass 0, the lower road plane 

qd[1] = road plane linear velocity 

>	Tr[1] := kinetic_energy( 'translation', M[us], qd[1] );

 

(4.2.1)

 

= kinetic energy of the the Mass 1, the middle tire platform 

qd[2] = Tire plane linear velocity 

>	Tr[2] := kinetic_energy( 'translation', M[s], qd[2] );

 

(4.2.2)

 

=  kinetic energy of the the Mass 2, the top car platform 

qd[3] = Tire plane linear velocity 

= Total Kinetic Energy of the system 

>	T[T] := Tr[1] + Tr[2] ;

 

(4.2.3)

 

>	T[T] := collect( T[T], diff );

 

(4.2.4)

 

Additional Insight: Eliminating Gravity Terms 

>	xeq := [x[eq1](t), x[eq2](t)];

 

(6.1.1)

 

>	seq(  q[i] = xeq[i] , i=1..Nq );

 

(6.1.2)

 

>	EOM_NoGrav := simplify( subs( [seq(  q[i] = xeq[i] , i=1..Nq), seq( diff( q[i], t) = 0 , i=1..Nq),F[c]=0,z[r](t)=0] ,  EOM_New  ) );

 

(6.1.3)

 

>	xeq := solve( EOM_NoGrav, xeq );

 

(6.1.4)

 

>	x[eq1] := -(M[s]+M[us])*g/K[us] ;                                                                                                     x[eq2] := -M[s]*g/K[s]-(M[s]+M[us])*g/K[us] ;

 

 

	(6.1.5)

 

>

z_var := [ z[us](t), z[s](t) ];                                                                                              subs_zx := { x[1](t) = z_var[1]+x[eq1] ,     x[2](t) = z_var[2]+x[eq2] }; subs_zxd := { seq( diff( q[i], t) = diff( z_var[i], t) , i=1..Nq ) };                                                       subs_zxdd := { seq( diff( q[i], t$2 ) =  diff( z_var[i], t$2), i=1..Nq ) };

 

 

 

 

	(6.1.6)

 

>	EOM_noG := subs( subs_zxd, subs( subs_zxdd, EOM_orig ) );

 

(6.1.7)

 

>	EOM_noG := subs( subs_zx, EOM_noG ) ;

 

(6.1.8)

 

>	EOM_noG_S := collect( EOM_noG,{  K[s], K[us]},distributed,factor);

 

(6.1.9)

 

>

subs_Zq := [ z_var[1]-z[r](t)=Z[1], z_var[2]-z_var[1]=Z[2], z_var[1]-z_var[2]=-Z[2]   ];                                          subs_Zqd := { seq( diff ( z_var, t )[i] = Z[i+Nq], i=1..Nq ) };                                                          subs_Zqdd := { seq( diff( z_var[i], t$2 ) =  Zd[i+Nq], i=1..Nq ) };

 

 

 

	(6.1.10)

 

>	EOM_states_noG := subs( subs_Zq, subs( subs_Zqdd, EOM_noG_S ) );

 

(6.1.11)

 

>	EOM_states_noG := subs( subs_Zqd, subs_U, EOM_states_noG );

 

(6.1.12)

 

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:

 

Here, Fi shows a linear system 

>	'F[i]' = Fi;

 

(6.2.1)

 

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

only for pretty print 

>	subs_Xq_rev := { seq( X[i] = q1[i+3], i=1..Nq ) }; subs_Xqd_rev := { seq( X[i+Nq] = qd[i], i=1..Nq ) };

 

 

	(7.1.1)

 

>	#subs_U_rev := { U[1] = V[m] }: subs_U_rev := { U[1] = ud,U[2] = F[c],U[3]=g};

 

(7.1.2)

 

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

 

(7.1.3)

 

Solution to the (Linear) EOM's 

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

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

 

(7.2.1)

 

>	assign( Xqdd_solset_ser );

 

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

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

 

(7.2.2)

 

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

 

(7.2.3)