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/*
* MIT No Attribution
*
* Copyright (C) 2010-2023 Joel Andersson, Joris Gillis, Moritz Diehl, KU Leuven.
*
* Permission is hereby granted, free of charge, to any person obtaining a copy of this
* software and associated documentation files (the "Software"), to deal in the Software
* without restriction, including without limitation the rights to use, copy, modify,
* merge, publish, distribute, sublicense, and/or sell copies of the Software, and to
* permit persons to whom the Software is furnished to do so.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED,
* INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A
* PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
* HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
* OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
* SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
*
*/
#include <casadi/casadi.hpp>
#include <string.h>
/** Solve a simple QP
We want to model a chain attached to two supports and hanging in between. Let us discretise
it with N mass points connected by N-1 springs. Each mass i has position (yi,zi), i=1,...,N.
The equilibrium point of the system minimises the potential energy.
The potential energy of each spring is
Vi=D_i/2 * ((y_i-y_{i+1})^2 + (z_i-z_{i+1})^2)
The gravitational potential energy of each mass is
Vg_i = m_i*g0*z_i
The total potential energy is thus given by:
Vchain(y,z) = 1/2*sum{i=1,...,N-1} D_i ((y_i-y_{i+1})^2+(z_i-z_{i+1})^2) + g0 * sum{i=1,...,N} m_i * z_i
where y=[y_1,...,y_N] and z=[z_1,...,z_N]
We wish to solve
minimize{y,z} Vchain(y, z)
Subject to the piecewise linear ground constraints:
z_i >= zin
z_i - 0.1*y_i >= 0.5
Joel Andersson, 2015
*/
using namespace casadi;
int main(int argc, char* argv[]) {
// Default options
std::string plugin_name = "qpoases";
bool schur = false;
bool large = false;
// Read options, first is plugin name
if (argc>1) plugin_name = argv[1];
for (int i=2; i<argc; ++i) {
if (strcmp(argv[i], "schur")==0) schur = true;
if (strcmp(argv[i], "large")==0) large = true;
}
// Constants
int N = large ? 200 : 40;
double m_i = 40.0/N;
double D_i = 70.0*N;
double g0 = 9.81;
//double zmin = -inf; // unbounded
double zmin = 0.5; // ground
// Objective function
SX Vchain = 0;
// Variables
std::vector<SX> x;
// Variable bounds
std::vector<double> lbx, ubx;
// Constraints
std::vector<SX> g;
// Constraint bounds
std::vector<double> lbg, ubg;
// Loop over all chain elements
SX y_prev, z_prev, y_i, z_i;
for (int i=1; i<N+1; ++i) {
// Previous point
if (i>1) {
y_prev = y_i;
z_prev = z_i;
}
// Create variables for the (y_i, z_i) coordinates
y_i = SX::sym("y_" + str(i));
z_i = SX::sym("z_" + str(i));
// Add to the list of variables
x.push_back(y_i);
x.push_back(z_i);
if (i==1) {
lbx.push_back(-2);
ubx.push_back(-2);
lbx.push_back( 1);
ubx.push_back( 1);
} else if (i==N) {
lbx.push_back( 2);
ubx.push_back( 2);
lbx.push_back( 1);
ubx.push_back( 1);
} else {
lbx.push_back(-inf);
ubx.push_back( inf);
lbx.push_back(zmin);
ubx.push_back( inf);
}
// Spring potential
if (i>1) {
Vchain += D_i/2*(sq(y_prev-y_i) + sq(z_prev-z_i));
}
// Graviational potential
Vchain += g0 * m_i * z_i;
// Slanted ground constraints
g.push_back(z_i - 0.1*y_i);
lbg.push_back( 0.5);
ubg.push_back( inf);
}
// Formulate QP
SXDict qp = {{"x", vertcat(x)}, {"f", Vchain}, {"g", vertcat(g)}};
// Solver specific options
Dict solver_options;
if (plugin_name == "qpoases") {
solver_options["sparse"] = true;
solver_options["schur"] = schur;
solver_options["print_time"] = true;
}
// Create solver instance
Function solver = qpsol("solver", plugin_name, qp, solver_options);
// Get the optimal solution
DMDict arg = {{"lbx", lbx},
{"ubx", ubx},
{"lbg", lbg},
{"ubg", ubg}};
DMDict res = solver(arg);
DM x_opt = res["x"];
double f_opt(res["f"]);
std::cout << "f_opt = " << f_opt << std::endl;
// Retrieve the result
DM y_opt = x_opt(Slice(0, x_opt.size1(), 2));
DM z_opt = x_opt(Slice(1, x_opt.size1(), 2));
// Create Matlab script to plot the solution
std::ofstream file;
std::string filename = "chain_qp_results.m";
file.open(filename.c_str());
file << "% Results file from " __FILE__ << std::endl;
file << "% Generated " __DATE__ " at " __TIME__ << std::endl;
file << std::endl;
file << "t = linspace(-2,2," << N << ");"<< std::endl;
file << "f_opt = " << f_opt << ";" << std::endl;
file << "x_opt = " << x_opt << ";" << std::endl;
file << "y_opt = " << y_opt << ";" << std::endl;
file << "z_opt = " << z_opt << ";" << std::endl;
// Finalize the results file
file << std::endl;
file << "% Plot the results" << std::endl;
file << "figure(1);" << std::endl;
file << "clf;" << std::endl;
file << "plot(y_opt, z_opt);" << std::endl;
file << "grid on;" << std::endl;
file << "xlabel('y [m]');" << std::endl;
file << "ylabel('z [m]');" << std::endl;
file.close();
std::cout << "Results saved to \"" << filename << "\"" << std::endl;
return 0;
}
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