% MotionGenesis file: MGSolveNonlinearEqnsContinuouslyC.txt
% Copyright (c) 2024 Motion Genesis LLC. All rights reserved.
% Purpose: Continuous solution to the nonlinear algebraic equations:
%          x = r*cos(theta)  where x = cos(1.2*t)
%          y = r*sin(theta)  where y = sin(t)
% Related: MGSolveNonlinearEqnContinuouslyA.txt
%          MGSolveNonlinearEqnsContinuouslyB.txt
%          MGSolveNonlinearEqnsContinuouslyC.txt
%          MGFigureEightLinkageInverseKinematics.txt
%-------------------------------------------------------------------------------
Specified x', y'
SetDt( x = cos(1.2*t) )
SetDt( y = sin(t) )
%-------------------------------------------------------------------------------
Variable  r', theta'
eqnToSolve[1] = x - r*cos(theta)
eqnToSolve[2] = y - r*sin(theta)
%-------------------------------------------------------------------------------
%   Differentiate eqnToSolve to form equations that are _linear_ in r', theta'.
eqnDt = Dt( eqnToSolve )
%-------------------------------------------------------------------------------
%   Solve eqnDt = 0 to form an ODE (ordinary differential equation) for r' and theta'
Solve( eqnDt = 0,  r', theta' )
%-------------------------------------------------------------------------------
%   Form a set of equations that govern the values of r and theta at t = 0.
eqnInitial = Evaluate( eqnToSolve, t = 0 )
%-------------------------------------------------------------------------------
%   Solve the nonlinear equations at t = 0 to set initial values for qA and qB.
%   Since the equations are nonlinear, provide a guess for qA and qB
SolveSetInput( eqnInitial = 0,  r = 2 meters,  theta = 10 degrees )
%-------------------------------------------------------------------------------
%   List output quantities (for subsequent ODE command).
Output  t seconds, x meters,  y meters, r meters,  theta degrees
%--------------------------------------------------------------------
%   Set numerical integration parameters and solve ODEs.
Input  tFinal = 7 sec,  tStep = 0.02 sec,  absError = 1.0E-07
ODE()  MGSolveNonlinearEqnsContinuouslyC
%-------------------------------------------------------------------------------
%   Optional: Plot results.
% Plot MGSolveNonlinearEqnsContinuouslyC.1 [2, 3]   % Graph y vs. x.
% Plot MGSolveNonlinearEqnsContinuouslyC.1 [1, 5]   % Graph theta vs. time.
%-------------------------------------------------------------------------------
%   Record input together with responses.
Save  MGSolveNonlinearEqnsContinuouslyC.html
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