diff --git a/python/IK_manipulator_manipulability.py b/python/IK_manipulator_manipulability.py
new file mode 100644
index 0000000000000000000000000000000000000000..923d6f47e8a29572420f2f6bf154482bdc07543c
--- /dev/null
+++ b/python/IK_manipulator_manipulability.py
@@ -0,0 +1,381 @@
+'''
+Inverse kinematics with visualization of manipulability
+
+Copyright (c) 2024 Idiap Research Institute <https://www.idiap.ch/>
+Written by Sylvain Calinon <https://calinon.ch>
+
+This file is part of RCFS <https://robotics-codes-from-scratch.github.io/>
+License: GPLv3
+'''
+
+import numpy as np
+import random
+import matplotlib
+import matplotlib.pyplot as plt
+import math as m
+import scipy
+
+
+# Logarithmic map for R^2 x S^1 manifold
+def logmap(f, f0):
+ diff = np.zeros(3)
+ diff[:2] = f[:2] - f0[:2] # Position residual
+ diff[2] = np.imag(np.log(np.exp(f0[-1]*1j).conj().T * np.exp(f[-1]*1j).T)).conj() # Orientation residual
+ return diff
+
+# Forward kinematics for end-effector (in robot coordinate system)
+def fkin(x, param):
+ L = np.tril(np.ones([param.nbVarX, param.nbVarX]))
+ f = np.stack([
+ param.l @ np.cos(L @ x),
+ param.l @ np.sin(L @ x),
+ np.mod(np.sum(x,0)+np.pi, 2*np.pi) - np.pi
+ ]) # f1,f2,f3, where f3 is the orientation (single Euler angle for planar robot)
+ return f
+
+# Forward kinematics for all joints (in robot coordinate system)
+def fkin0(x, param):
+ L = np.tril(np.ones([param.nbVarX, param.nbVarX]))
+ f = np.vstack([
+ L @ np.diag(param.l) @ np.cos(L @ x),
+ L @ np.diag(param.l) @ np.sin(L @ x)
+ ])
+ f = np.hstack([np.zeros([2,1]), f])
+ 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
+
+## Parameters
+# ===============================
+
+param = lambda: None # Lazy way to define an empty class in python
+param.dt = 1e-2 # Time step length
+param.nbData = 50 # Number of datapoints
+param.nbVarX = 3 # State space dimension (x1,x2,x3)
+param.nbVarU = 3 # Control space dimension (dx1,dx2,dx3)
+param.nbVarF = 3 # Objective function dimension (position and orientation of the end-effector)
+param.l = [2, 2, 1] # Robot links lengths
+
+fig, ax = plt.subplots()
+
+
+fh = np.array([3, 1, -np.pi/2]) # Desired target for the end-effector (position and orientation)
+x = -np.ones(param.nbVarX) * np.pi / param.nbVarX # Initial robot pose
+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)
+ J = Jkin(x, param) # Jacobian (for end-effector)
+# x += np.linalg.pinv(J) @ (fh - f) * 10 * param.dt # Update state
+ x += np.linalg.pinv(J) @ logmap(fh, f) * 10 * param.dt # Update state
+
+ 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
+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
+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)
+for count1 in range(2 ** param.nbVarX):
+ for count2 in range(len(bin(count1)) - 2):
+ vertex[-count2-1] = int(str(bin(count1)[-count2-1]))
+ cube[count1] = vertex
+
+# 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
+polytope = np.zeros((2 ** param.nbVarX,2))
+for count in range(2 ** param.nbVarX):
+ polytope[count] = J @ cube[count] + center
+
+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--')
+
+
+def norm(vec, coeff, exp):
+ terms = abs(vec/coeff)**exp
+ norm = sum(terms) ** (1/exp)
+ return norm
+
+def sample(npoints, coeff, exp):
+ vecs = np.random.rand(npoints, param.nbVarX) * 2 - 1
+ vecs *= size
+
+ for count in range(len(vecs)):
+ 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
+
+ ell_jvlim = sample(num_iter, coeff, 2)
+
+ ell_tvlim = np.zeros((num_iter,2))
+
+ for count in range(len(ell_jvlim)):
+ ell_tvlim[count] = J @ ell_jvlim[count] + center
+
+ ell_x, ell_y = ell_tvlim.T
+
+ A = np.diag(coeff ** 2)
+ Q = J @ A @ J.T
+ eigenvals, eigenvecs = np.linalg.eig(Q)
+ # 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
+
+
+ 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()