diff --git a/python/IK_manipulator_manipulability.py b/python/IK_manipulator_manipulability.py
index ba110964de82621966d665a6aa733d3dadfc7b45..e971b29103af250bfb3946b04f4aedca6b5ead89 100644
--- a/python/IK_manipulator_manipulability.py
+++ b/python/IK_manipulator_manipulability.py
@@ -73,7 +73,6 @@ x[0] = x[0] + np.pi
## Inverse kinematics (IK)
# ===============================
-
ax.scatter(fh[0], fh[1], color='r', marker='.', s=10**2) #Plot target
for t in range(param.nbData):
f = fkin(x, param) # Forward kinematics (for end-effector)
@@ -83,72 +82,15 @@ for t in range(param.nbData):
f_rob = fkin0(x, param) # Forward kinematics (for all articulations, including end-effector)
ax.plot(f_rob[0,:], f_rob[1,:], color=str(1-t/param.nbData), linewidth=2) # Plot robot
-
-
### MANIPULABILITY ###
-
J = J[:2,:]
center = np.array([f[0], f[1]]) # end-effector position
+print(f"Center: {center}")
length, width, height = 1.8,1.5,1 # max joint velocities
size = np.array([length, width, height])
-refell = 130 * np.identity(2) # reference ellipsoid
-
-# Choice of the Jacobian matrix
-J1 = False
-J2 = False
-J3 = False
-J4 = False
-diffJac = [J1, J2, J3, J4]
-
-
-# 1. Robot manipulator
-if J1 == True:
- theta = 5*m.pi/6
- U = np.array([[m.cos(theta), -m.sin(theta)],[m.sin(theta), m.cos(theta)]])
- J = U.T @ J
- print(J)
-
-# 2. Bounded joint-space
-if J2 == True:
- jminlim = -np.ones(param.nbVarX)
- jmaxlim = np.ones(param.nbVarX)
- J = np.diag(1 - np.heaviside(x - jminlim,0)*np.heaviside(jmaxlim - x, 0))[:2,:]
- print(J)
-
-
-#Â 3. Bounded task-space
-if J3 == True:
- tminlim = -np.ones(2)
- tmaxlim = np.ones(2)
- J = np.diag(1 - np.heaviside(f[:2] - tminlim,0)*np.heaviside(tmaxlim - f[:2], 0)) @ J
- print(J)
-
-# 4. Object boundaries
-if J4 == True:
- theta = m.pi/4
- U = np.array([[m.cos(theta), -m.sin(theta)],[m.sin(theta), m.cos(theta)]])
- tminlim = -np.ones(2)
- tmaxlim = 2*np.ones(2)
- J = np.diag(1 - np.heaviside(U.T@(f[:2] - fh[:2]) - tminlim,0)*np.heaviside(tmaxlim - (U.T@(f[:2] - fh[:2])), 0)) @ J
- print(J)
-
-
-
-# Boundaries in joint-velocity space
-
-# 1. Rectangular cuboid
-showedges = False # Shows the mapping of the cube's edges
-
-# 2. Ellipse
-ellBound = True
-# 3. Superellipsoid
-superBound = False
-superVolume = False # Returns the fraction of the rectangular cuboid's volume covered by the superellipsoid
-
-
-# 1. Rectangular cuboid
+# initialize a rectangular cuboid for polytope
cube = np.zeros((2 ** param.nbVarX, param.nbVarX))
vertex = np.zeros(param.nbVarX)
# These two loops store the numbers 0 to 7 in binary (which can be seen as the coordinates of a cube)
@@ -159,12 +101,10 @@ for count1 in range(2 ** param.nbVarX):
# Rescaling so that the center of the cube is located at the origin
cube = cube * 2 - 1
-
-
for i in range(len(size)):
cube[:,i] = cube[:,i] * size[i]
-# Computation of the manipulability polytope
+# Computation of the manipulability polytope (blue)
polytope = np.zeros((2 ** param.nbVarX,2))
for count in range(2 ** param.nbVarX):
polytope[count] = J @ cube[count] + center
@@ -173,17 +113,13 @@ xpoints = polytope[:,0]
ypoints = polytope[:,1]
polytope = np.array([xpoints, ypoints]).T
-if not any(diffJac) == True:
- hull = scipy.spatial.ConvexHull(polytope)
-
- # vertices of the covex hull (might come in handy)
- vertices = np.zeros((len(hull.vertices),2))
- for i in range(len(hull.vertices)):
- vertices[i] = polytope[hull.vertices[i]]
- cube_norms = np.linalg.norm(vertices, axis = 1)
-
- for simplex in hull.simplices:
- plt.plot(polytope[simplex, 0], polytope[simplex, 1], 'b--')
+hull = scipy.spatial.ConvexHull(polytope)
+vertices = np.zeros((len(hull.vertices),2))
+for i in range(len(hull.vertices)):
+ vertices[i] = polytope[hull.vertices[i]]
+ax.plot(xpoints, ypoints, "kx")
+for simplex in hull.simplices:
+ plt.plot(polytope[simplex, 0], polytope[simplex, 1], 'b--')
def norm(vec, coeff, exp):
@@ -199,189 +135,26 @@ def sample(npoints, coeff, exp):
vecs[count] = vecs[count] / norm(vecs[count], coeff, exp)
return vecs
-# 2. Ellipsoid
-if ellBound == True:
- num_iter = 1000
- # coeff = np.array([1,1,1]) # these are the dimensions of the superellipsoid in joint-velocity space
- coeff = size # if one wants the superellipsoid to be contained in the cuboid
- exp = 2
+# plot the manipulability ellipsoid (red)
+num_iter = 1000
+coeff = np.array([1,1,1]) # these are the dimensions of the superellipsoid in joint-velocity space
+# coeff = size # if one wants the superellipsoid to be contained in the cuboid
- ell_jvlim = sample(num_iter, coeff, 2)
-
- ell_tvlim = np.zeros((num_iter,2))
+ell_jvlim = sample(num_iter, coeff, 2)
- for count in range(len(ell_jvlim)):
- ell_tvlim[count] = J @ ell_jvlim[count] + center
+ell_tvlim = np.zeros((num_iter,2))
- ell_x, ell_y = ell_tvlim.T
+for count in range(len(ell_jvlim)):
+ ell_tvlim[count] = J @ ell_jvlim[count] + center
- A = np.diag(coeff ** 2)
- Q = J @ A @ J.T
- eigenvals, eigenvecs = np.linalg.eig(Q)
-
- # Sort Eigenvalues and EigenVectors
- idx = eigenvals.argsort()[::-1]
- eigenvals = eigenvals[idx]
- eigenvecs = eigenvecs[idx]
+ell_x, ell_y = ell_tvlim.T
- # the sqrt of the eigenvalues give the length of the semi-axes
- print(f"Ellipsoid eigenvalues: {eigenvals}")
- vec1, vec2 = eigenvecs.T
- vec1 = vec1 * m.sqrt(eigenvals[0]) + center
- vec2 = vec2 * m.sqrt(eigenvals[1]) + center
+polytope = np.array([ell_x, ell_y]).T
+hull = scipy.spatial.ConvexHull(polytope)
+for simplex in hull.simplices:
+ plt.plot(polytope[simplex, 0], polytope[simplex, 1], 'r--')
- if not any(diffJac) == True:
- polytope = np.array([ell_x, ell_y]).T
- hull = scipy.spatial.ConvexHull(polytope)
- # vertices of the covex hull (might come in handy)
- #vertex = np.zeros((len(hull.vertices),2))
- #for i in range(len(hull.vertices)):
- # vertex[i] = polytope[hull.vertices[i]]
- #ax.plot(vertex[:,0], vertex[:,1], "gv")
-
- for simplex in hull.simplices:
-
- plt.plot(polytope[simplex, 0], polytope[simplex, 1], 'r--')
-
-
-
-# 3. Superellipsoid (rigorously this is not the most general form of a superellipsoid)
-if superBound == True:
- num_iter = 1000
- # coeff = np.array([1,1,1]) # these are the dimensions of the superellipsoid in joint-velocity space
- coeff = size # if one wants the superellipsoid to be contained in the cuboid
- exp = 4 # exp = 2 for an ellipse, exp = 4 for squircle, exp --> infty for rectangular cuboid
-
- if superVolume == True:
- vol = scipy.special.gamma(1/exp + 1)**param.nbVarX/scipy.special.gamma(param.nbVarX/exp + 1)
- print(f"fraction of the rectangular cuboid's volume: {vol}")
-
- jvlim = sample(num_iter, coeff, exp)
-
- tvlim = np.zeros((num_iter,2))
-
- for count in range(len(jvlim)):
- tvlim[count] = J @ jvlim[count] + center
-
-
- xpoints, ypoints = tvlim.T
-
- # Idea: approximate whatever shape I get with an ellipsoid, so that the reasoning on the eigenvalues apply!
- # Note: it does not just give the same ellipsoid as if exp = 2
-
- tvmax = tvlim[np.argmax(np.linalg.norm(tvlim-center, axis = 1))]
- cov_mat = np.cov(tvlim.T)
- eigenvals, eigenvecs = np.linalg.eig(cov_mat)
- idx = eigenvals.argsort()[::-1]
- eigenvals = eigenvals[idx]
- eigenvecs = eigenvecs[:,idx]
- eigenvecs = eigenvecs * np.sqrt(eigenvals)
- ratio = np.linalg.norm(tvmax-center)/m.sqrt(eigenvals[0])
- eigenvecs *= ratio
-
- vec1, vec2 = eigenvecs.T[0],eigenvecs.T[1]
-
- '''
- # Manipulability matrix
- Q = eigenvecs @ np.array([[eigenvals[0],0],[0,eigenvals[1]]]) @ np.linalg.inv(eigenvecs)
-
- # Riemannian distance
- A = np.linalg.inv(scipy.linalg.sqrtm(refell)) @ Q @ np.linalg.inv(scipy.linalg.sqrtm(refell))
- d = np.linalg.norm(scipy.linalg.logm(A))
- print(f"Riemannian distance: {d}")
- '''
-
- print(f"Superellipsoid eigenvalues: {(ratio * np.sqrt(eigenvals))**2}")
-
- phi = np.linspace(0, 2*m.pi,200)
- x = np.zeros((len(phi),2))
-
- for i in range(len(phi)):
- x[i] = center + vec1 * m.cos(phi[i]) + vec2 * m.sin(phi[i])
-
- super_norms = np.linalg.norm(x,axis = 1)
-
- if not any(diffJac) == True:
- ax.plot(x[:,0], x[:,1], "g1", label = "superellipsoid")
- vec1 += center
- vec2 += center
- ax.plot([center[0], tvmax[0]], [center[1],tvmax[1]])
- ax.plot([center[0], vec1[0]], [center[1],vec1[1]])
- ax.plot([center[0], vec2[0]], [center[1],vec2[1]])
-
-
-# Plots
-showhull = True # to show the convex hull of the superellipsoid
-showpoints = False # to show the image of all the sampled points
-
-
-if showhull == True and not any(diffJac) == True:
- polytope = np.array([xpoints, ypoints]).T
- hull = scipy.spatial.ConvexHull(polytope)
-
- # vertices of the covex hull (might come in handy)
- #vertex = np.zeros((len(hull.vertices),2))
- #for i in range(len(hull.vertices)):
- # vertex[i] = polytope[hull.vertices[i]]
- #ax.plot(vertex[:,0], vertex[:,1], "gv")
-
- for simplex in hull.simplices:
-
- plt.plot(polytope[simplex, 0], polytope[simplex, 1], 'g--')
-
-if showpoints == True:
- ax.plot(xpoints, ypoints, "kx")
-
-#fig = plt.figure()
-#ax2 = fig.add_subplot(projection='3d')
-
-#ax2.scatter(cube[:,0], cube[:,1], cube[:,2], c = "blue", label = "rectangular cuboid")
-
-#if ellBound == True:
-# ax2.scatter(ell_jvlim[:,0], ell_jvlim[:,1], ell_jvlim[:,2], c = "red", label = "ellipsoid")
-#
-#if superBound == True:
-# ax2.scatter(jvlim[:,0], jvlim[:,1], jvlim[:,2], c = "green", label = "superellipsoid")
-
-#legend = ax2.legend(loc='upper right')
-
-
-#if showedges == True:
-# num_points = 50
-# jvlim, tvlim = np.zeros((num_points,3)), np.zeros((num_points,2))
-#
-# edges = np.vstack((np.unique(cube[:,:2], axis = 0), np.unique(cube[:,1:3], axis = 0), np.unique(cube[:,0:3:2], axis = 0)))
-# for edge in edges[:4]:
-# for count in range(num_points):
-# z = (random.random() * 2 - 1) * height
-# jvlim[count] = np.array([edge[0],edge[1],z])
-# tvlim[count] = J @ jvlim[count] + center
-# ax2.scatter(jvlim[:,0], jvlim[:,1], jvlim[:,2], c = "blue")
-# ax.plot(tvlim[:,0], tvlim[:,1], "bx")
-
-# for edge in edges[4:8]:
-# for count in range(num_points):
-# x = (random.random() * 2 - 1) * length
-# jvlim[count] = np.array([x,edge[0],edge[1]])
-# tvlim[count] = J @ jvlim[count] + center
-# ax2.scatter(jvlim[:,0], jvlim[:,1], jvlim[:,2], c = "red")
-# ax.plot(tvlim[:,0], tvlim[:,1], "rx")
-#
-# for edge in edges[8:]:
-# for count in range(num_points):
-# y = (random.random() * 2 - 1) * width
-# jvlim[count] = np.array([edge[0],y,edge[1]])
-# tvlim[count] = J @ jvlim[count] + center
-# ax2.scatter(jvlim[:,0], jvlim[:,1], jvlim[:,2], c = "green")
-# ax.plot(tvlim[:,0], tvlim[:,1], "gx")
-#
-
-
-#ax.axis('off')
ax.axis('equal')
-#ax2.axis('equal')
-#plt.title(f"Length: {length}, width: {width}, height: {height}, p = {exp}")
-
plt.show()
diff --git a/python/LQR_probabilistic.py b/python/LQR_probabilistic.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d06ed095f193c65c0df724f0aa2ba2f2fa84988
--- /dev/null
+++ b/python/LQR_probabilistic.py
@@ -0,0 +1,66 @@
+import numpy as np
+from math import factorial
+import matplotlib.pyplot as plt
+from scipy.linalg import solve_continuous_are
+from scipy.stats import multivariate_normal
+
+# Parameters
+# ===============================
+param = lambda: None # Lazy way to define an empty class in python
+param.dt = 1e-2 # Time step length
+param.nbDeriv = 2 # Order of the dynamical system
+param.nbVarPos = 2 # Number of position variable
+param.nbVar = param.nbVarPos * param.nbDeriv # Dimension of state vector
+param.nbData = 100 # Number of datapoints
+param.rfactor = 1e-7 # Control weight term
+
+R = np.eye((param.nbData-1) * param.nbVarPos) * param.rfactor # Control cost matrix
+Q = np.zeros((param.nbVar * param.nbData, param.nbVar * param.nbData)) # Task precision for augmented state
+
+xd = np.zeros([param.nbVar, param.nbData])
+target = np.random.uniform(size=param.nbVarPos)
+xd[:, param.nbData-1] = np.concatenate((target, np.zeros(param.nbVarPos)))
+xd = xd.T.flatten()
+
+Q[param.nbVar*(param.nbData-1):param.nbVar*param.nbData,param.nbVar*(param.nbData-1):param.nbVar*param.nbData] = 10.0 * np.eye(param.nbVar)
+
+A1d = np.zeros((param.nbDeriv,param.nbDeriv))
+B1d = np.zeros((param.nbDeriv,1))
+
+for i in range(param.nbDeriv):
+ A1d += np.diag( np.ones(param.nbDeriv-i), i ) * param.dt**i * 1/factorial(i)
+ B1d[param.nbDeriv-i-1] = param.dt**(i+1) * 1/factorial(i+1)
+
+A = np.eye(param.nbVar)
+A[:param.nbVar, :param.nbVar] = np.kron(A1d, np.identity(param.nbVarPos))
+B = np.zeros((param.nbVar, param.nbVarPos))
+B[:param.nbVar] = np.kron(B1d, np.identity(param.nbVarPos))
+
+Su = np.zeros((param.nbVar*param.nbData, param.nbVarPos * (param.nbData-1)))
+Sx = np.kron(np.ones((param.nbData,1)), np.eye(param.nbVar,param.nbVar))
+
+M = B
+for i in range(1,param.nbData):
+ Sx[i*param.nbVar:param.nbData*param.nbVar,:] = np.dot(Sx[i*param.nbVar:param.nbData*param.nbVar,:], A)
+ Su[param.nbVar*i:param.nbVar*i+M.shape[0],0:M.shape[1]] = M
+ M = np.hstack((np.dot(A,M),B)) # [0,nb_state_var-1]
+
+
+x0 = np.random.uniform(size=param.nbVar)
+mean = Sx @ x0 + Su @ np.linalg.inv(Su.T @ Q @ Su + R) @ Su.T @ Q @ (xd - Sx @ x0)
+cov = Su @ (Su.T @ Q @ Su + R) @ Su.T
+
+
+trajectories = multivariate_normal.rvs(mean=mean, cov=cov, size=100)
+
+plt.figure()
+for k in range(100):
+ trajectory = trajectories[k, :]
+ trajectory = trajectory.reshape((param.nbData, param.nbVar))
+ plt.plot(trajectory[:, 0], trajectory[:, 1], alpha=0.5)
+
+plt.scatter(x0[0], x0[1], color='green', label='Initial State')
+plt.scatter(target[0], target[1], color='red', label='Target State')
+plt.legend()
+plt.grid()
+plt.show()