diff --git a/python/iLQR_bimanual_manipulability.py b/python/iLQR_bimanual_manipulability.py
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-'''
- iLQR applied to a planar bimanual robot problem with a cost on tracking a desired
-
- Copyright (c) 2021 Idiap Research Institute, http://www.idiap.ch/
- Written by Boyang Ti <https://www.idiap.ch/~bti/> and
- Sylvain Calinon <https://calinon.ch>
-
- This file is part of RCFS.
-
- RCFS is free software: you can redistribute it and/or modify
- it under the terms of the GNU General Public License version 3 as
- published by the Free Software Foundation.
-
- RCFS is distributed in the hope that it will be useful,
- but WITHOUT ANY WARRANTY; without even the implied warranty of
- MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- GNU General Public License for more details.
-
- You should have received a copy of the GNU General Public License
- along with RCFS. If not, see <http://www.gnu.org/licenses/>.
-'''
-
-import numpy as np
-import matplotlib.pyplot as plt
-import matplotlib.patches as patches
-import scipy.linalg
-from scipy.linalg import fractional_matrix_power
-
-import numpy.matlib
-# Helper functions
-# ===============================
-
-# Forward kinematics for end-effector (in robot coordinate system)
-def fkin(x, param):
- L = np.tril(np.ones(3))
- f = np.vstack((
- param.l[0:3].T @ np.cos(L @ x[0:3]),
- param.l[0:3].T @ np.sin(L @ x[0:3]),
- param.l[[0, 3, 4]].T @ np.cos(L @ x[[0, 3, 4]]),
- param.l[[0, 3, 4]].T @ np.sin(L @ x[[0, 3, 4]])
- )) # f1,f2
- return f
-
-# Forward kinematics for end-effector (in robot coordinate system)
-def fkin0(x, param):
- L = np.tril(np.ones(3))
- fl = np.vstack((
- L @ np.diag(param.l[0:3].flatten()) @ np.cos(L @ x[0:3]),
- L @ np.diag(param.l[0:3].flatten()) @ np.sin(L @ x[0:3])
- ))
- fr = np.vstack((
- L @ np.diag(param.l[[0, 3, 4]].flatten()) @ np.cos(L @ x[[0, 3, 4]]),
- L @ np.diag(param.l[[0, 3, 4]].flatten()) @ np.sin(L @ x[[0, 3, 4]])
- ))
- f = np.hstack((fl[:, ::-1], np.zeros([2, 1]), fr))
- return f
-
-# Forward kinematics for center of mass of a bimanual robot (in robot coordinate system)
-def fkin_CoM(x, param):
- L = np.tril(np.ones(3))
- f = np.vstack((param.l[0:3].T @ L @ np.cos(L @ x[0:3]) + param.l[[0, 3, 4]].T @ L @ np.cos(L @ x[[0, 3, 4]]),
- param.l[0:3].T @ L @ np.sin(L @ x[0:3]) + param.l[[0, 3, 4]].T @ L @ np.sin(L @ x[[0, 3, 4]]))) / 6
-
- return f
-
-# Jacobian with analytical computation (for single time step)
-def Jkin(x, param):
- L = np.tril(np.ones([param.nbVarX, param.nbVarX]))
- J = np.vstack([
- -np.sin(L @ x).T @ np.diag(param.l) @ L,
- np.cos(L @ x).T @ np.diag(param.l) @ L,
- np.ones([1, param.nbVarX])
- ])
- return J
-
-# Jacobian for center of mass of a robot (in robot coordinate system)
-def Jkin_CoM(x, param):
- L = np.tril(np.ones(3))
- Jl = np.vstack((-np.sin(L @ x[0:3].reshape(-1, 1)).T @ L @ np.diag((param.l[0:3].T @ L).flatten()),
- np.cos(L @ x[0:3].reshape(-1, 1)).T @ L @ np.diag((param.l[0:3].T @ L).flatten()))) / 6
- Jr = np.vstack((-np.sin(L @ x[[0, 3, 4]].reshape(-1, 1)).T @ L @ np.diag((param.l[[0, 3, 4]].T @ L).flatten()),
- np.cos(L @ x[[0, 3, 4]].reshape(-1, 1)).T @ L @ np.diag((param.l[[0, 3, 4]].T @ L).flatten()))) / 6
- J = np.hstack(((Jl[:, 0] + Jr[:, 0]).reshape(-1, 1), Jl[:, 1:], Jr[:, 1:]))
- return J
-
-def rman(x, param):
- G = fractional_matrix_power(param.MuS, -0.5)
- f = np.zeros((3, np.size(x, 1)))
- for i in range(np.size(x, 1)):
- Jt = Jkin_CoM(x[:, i], param) # Jacobian for center of mass
- St = Jt @ Jt.T # manipulability matrix
-
- D, V = np.linalg.eig(G @ St @ G)
- E = V @ np.diag(np.log(D)) @ np.linalg.pinv(V)
-
- E = np.tril(E) * (np.eye(2) + np.tril(np.ones(2), -1) * np.sqrt(2))
- f[:, i] = E[np.where(E!=0)]
- return f
-
-# Jacobian for manipulability tracking with numerical computation
-def Jman_num(x, param):
- e = 1E-6
- X = np.matlib.repmat(x, 1, param.nbVarX)
- F1 = rman(X, param)
- F2 = rman(X + np.eye(param.nbVarX) * e, param)
- J = (F2 - F1) / e
- return J
-
-# Residuals f and Jacobians J for manipulability tracking
-# (c=f'*f is the cost, g=J'*f is the gradient, H=J'*J is the approximated Hessian)
-def f_manipulability(x, param):
- f = rman(x, param) # Residuals
- for t in range(np.size(x, 1)):
- if t == 0:
- J = Jman_num(x[:, t].reshape(-1, 1), param)
- else:
- J = scipy.linalg.block_diag(J, Jman_num(x[:, t].reshape(-1, 1), param)) # Jacobians
- return f, J
-
-## Parameters
-class Param:
- def __init__(self):
- self.dt = 1e0 # Time step length
- self.nbIter = 100 # Maximum number of iterations for iLQR
- self.nbPoints = 1 # Number of viapoints
- self.nbData = 10 # Number of datapoints
- self.nbVarX = 5 # State space dimension ([q1,q2,q3] for left arm, [q1,q4,q5] for right arm)
- self.nbVarU = self.nbVarX # Control space dimension ([dq1, dq2, dq3, dq4, dq5])
- self.nbVarF = 4 # Objective function dimension ([x1,x2] for left arm and [x3,x4] for right arm)
- self.l = np.ones((self.nbVarX, 1)) * 2. # Robot links lengths
- self.r = 1e-6 # Control weight term
- self.MuS = np.asarray([[3, 2], [2, 4]])
-param = Param()
-# Precision matrix
-Q = np.kron(np.identity(param.nbPoints), np.diag([0., 0., 0., 0.]))
-# Control weight matrix
-R = np.identity((param.nbData-1) * param.nbVarU) * param.r
-# Precision matrix for continuous CoM tracking
-Qc = np.kron(np.identity(param.nbData), np.diag([0., 0.]))
-
-# Time occurrence of viapoints
-tl = np.linspace(0, param.nbData, param.nbPoints+1)
-tl = np.rint(tl[1:]).astype(np.int64) - 1
-idx = np.array([i + np.arange(0, param.nbVarX, 1) for i in (tl*param.nbVarX)])
-
-# iLQR
-# ===============================
-
-u = np.zeros(param.nbVarU * (param.nbData-1)) # Initial control command
-x0 = np.array([np.pi/3, np.pi/2, np.pi/3, -np.pi/3, -np.pi/4]) # Initial state
-
-# Transfer matrices (for linear system as single integrator)
-Su0 = np.vstack([
- np.zeros([param.nbVarX, param.nbVarX*(param.nbData-1)]),
- np.tril(np.kron(np.ones([param.nbData-1, param.nbData-1]), np.eye(param.nbVarX) * param.dt))
-])
-Sx0 = np.kron(np.ones(param.nbData), np.identity(param.nbVarX)).T
-Su = Su0[idx.flatten()] # We remove the lines that are out of interest
-for i in range(param.nbIter):
- x = Su0 @ u + Sx0 @ x0 # System evolution
- x = x.reshape([param.nbVarX, param.nbData], order='F')
- f, J = f_manipulability(x[:, tl], param) # Residuals and Jacobians
- du = np.linalg.inv(Su.T @ J.T @ J @ Su + R) @ (-Su.T @ J.T @ f.flatten('F') - u * param.r) # Gauss-Newton update
- # Estimate step size with backtracking line search method
- alpha = 1
- cost0 = f.flatten('F').T @ f.flatten('F') + np.linalg.norm(u)**2 * param.r # Cost
- while True:
- utmp = u + du * alpha
- xtmp = Su0 @ utmp + Sx0 @ x0 # System evolution
- xtmp = xtmp.reshape([param.nbVarX, param.nbData], order='F')
- ftmp, _ = f_manipulability(xtmp[:, tl], param) # Residuals
- cost = ftmp.flatten('F').T @ ftmp.flatten('F') + np.linalg.norm(utmp)**2 * param.r # Cost
- if cost < cost0 or alpha < 1e-3:
- u = utmp
- print("Iteration {}, cost: {}".format(i,cost))
- break
- alpha /= 2
- if np.linalg.norm(du * alpha) < 1E-2:
- break # Stop iLQR iterations when solution is reached
-
-# Plots
-# ===============================
-plt.figure()
-plt.axis('off')
-plt.gca().set_aspect('equal', adjustable='box')
-
-fc = fkin_CoM(x, param)
-al = np.linspace(-np.pi, np.pi, 50)
-ax = plt.gca()
-
-# Plot desired manipulability ellipsoid
-D1, V1 = np.linalg.eig(param.MuS)
-D1 = np.diag(D1)
-R1 = np.real(V1@np.sqrt(D1+0j))
-msh1 = (R1 @ np.array([np.cos(al), np.sin(al)]) * 0.52).T + np.matlib.repmat(fc[:, -1].reshape(-1, 1), 1, 50).T
-p1 = patches.Polygon(msh1, closed=True, alpha=0.9)
-p1.set_facecolor([1, 0.7, 0.7])
-p1.set_edgecolor([1, 0.6, 0.6])
-ax.add_patch(p1)
-
-# Plot robot manipulability ellipsoid
-J = Jkin_CoM(x[:, -1], param)
-S = J @ J.T
-D2, V2 = np.linalg.eig(S)
-D2 = np.diag(D2)
-R2 = np.real(V2@np.sqrt(D2+0j))
-msh2 = (R2 @ np.array([np.cos(al), np.sin(al)]) * 0.5).T + np.matlib.repmat(fc[:, -1].reshape(-1, 1), 1, 50).T
-p2 = patches.Polygon(msh2, closed=True, alpha=0.9)
-p2.set_facecolor([0.4, 0.4, 0.4])
-p2.set_edgecolor([0.3, 0.3, 0.3])
-ax.add_patch(p2)
-
-# Plot CoM
-fc = fkin_CoM(x, param) # Forward kinematics for center of mass
-plt.plot(fc[0, 0], fc[1, 0], marker='o', markerfacecolor='none', markeredgewidth=4, markersize=10, markeredgecolor=[0.5, 0.5, 0.5]) # Plot CoM
-plt.plot(fc[0, tl[-1]], fc[1, tl[-1]], marker='o', markerfacecolor='none', markeredgewidth=4, markersize=10, markeredgecolor=[0.2, 0.2, 0.2]) # Plot CoM
-
-# Plot end-effectors paths
-f01 = fkin0(x[:, 0], param)
-f02 = fkin0(x[:, tl[0]], param)
-
-# Get points of interest
-f = fkin(x, param)
-plt.plot(f01[0, :], f01[1, :], c='black', linewidth=4, alpha=.2)
-plt.plot(f02[0, :], f02[1, :], c='black', linewidth=4, alpha=.4)
-
-plt.plot(f[0, :], f[1, :], c='black', marker='o', markevery=[0]+tl.tolist())
-plt.plot(f[2, :], f[3, :], c='black', marker='o', markevery=[0]+tl.tolist())
-
-
-plt.show()