% Blocks problem from homework problem 4.2
% used to demonstrate generalized speeds
overwrite on

% Set up problem
newtonian a
frames b,c
bodies d
points o
mass d=md
inertia d,i1,i2,i3,0,0,0
variables q{6}',u{6}'
% Note that q is for generalized coordinates
%       and u is for generalized speeds

% Rotation matrices
simprot(a,b,3,q1)
simprot(b,c,1,q2)
simprot(c,d,3,q3)
a_d=a_b*b_c*c_d

% Kinematic relationships and rotational kinematics for third approach
w_d_a>=u1*a1>+u2*a2>+u3*a3>
w_d_a_alt>=q1'*a3>+q2'*b1>+q3'*c3>
express(w_d_a_alt>,a)
diff>=w_d_a>-w_d_a_alt>
eqn[1]=dot(diff>,a1>)
eqn[2]=dot(diff>,a2>)
eqn[3]=dot(diff>,a3>)
solve(eqn,q1',q2',q3')
expand(q1')
expand(q2')
expand(q3')
q4'=u4
q5'=u5
q6'=u6
pause
alf_d_a>=dt(w_d_a>,d) % Via million dollar formula

% Translational kinematics for third approach
p_o_do>=q4*a1>+q5*a2>+q6*a3>
v_do_a>=dt(p_o_do>,a)
a_do_a>=dt(v_do_a>,a)

% Dynamics equations for second approach
zero=fr()+frstar()
expand(zero)

% Note that to solve these equations via numerical integration,
% you need to start with initial conditions for the following
% variables: q1->q6 and u1->u6. Then two sets of first order
% (rather than one set of second order) differential equations
% must be solved via numerical integration: q' = f(q,u) and
% u' = f(q,u). The dependence of the dynamical (i.e., u')
% differential equations on q remains due to a poor choice
% of generalized speeds, and in fact, the equations become
% extremely complicated. Furthermore, the kinematical
% (i.e., q') differential equations still depend on q, and
% even those equations are more complicated than when a better
% choice of generalized speeds is made.