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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.
#
#
# -*- coding: utf-8 -*-
"""
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
"""
from casadi import *
# Constants
N = 40
m_i = 40.0/N
D_i = 70.0*N
g0 = 9.81
#zmin = -inf # unbounded
zmin = 0.5 # ground
# Objective function
Vchain = 0
# Variables
x = []
# Variable bounds
lbx = []
ubx = []
# Constraints
g = []
# Constraint bounds
lbg = []
ubg = []
# Loop over all chain elements
for i in range(1, N+1):
# 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 += [y_i, z_i]
if i==1:
lbx += [-2., 1.]
ubx += [-2., 1.]
elif i==N:
lbx += [ 2., 1.]
ubx += [ 2., 1.]
else:
lbx += [-inf, zmin]
ubx += [ inf, inf]
# Spring potential
if i>1:
Vchain += D_i/2*((y_prev-y_i)**2 + (z_prev-z_i)**2)
# Graviational potential
Vchain += g0 * m_i * z_i
# Slanted ground constraints
g.append(z_i - 0.1*y_i)
lbg.append( 0.5)
ubg.append( inf)
# Formulate QP
qp = {'x':vertcat(*x), 'f':Vchain, 'g':vertcat(*g)}
# Solve with IPOPT
solver = qpsol('solver', 'qpoases', qp, {'sparse':True})
#solver = qpsol('solver', 'gurobi', qp)
#solver = nlpsol('solver', 'ipopt', qp)
# Get the optimal solution
sol = solver(lbx=lbx, ubx=ubx, lbg=lbg, ubg=ubg)
x_opt = sol['x']
f_opt = sol['f']
print('f_opt = ', f_opt)
# Retrieve the result
Y0 = x_opt[0::2]
Z0 = x_opt[1::2]
# Plot the result
import matplotlib.pyplot as plt
plt.plot(Y0,Z0,'o-')
ys = linspace(-2.,2.,100)
zs = 0.5 + 0.1*ys
plt.plot(ys,zs,'--')
plt.xlabel('y [m]')
plt.ylabel('z [m]')
plt.title('hanging chain QP')
plt.grid(True)
plt.legend(['Chain','z - 0.1y >= 0.5'],loc=9)
plt.show()
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