iLQR_manipulator3D.py added

python/iLQR_manipulator3D.py

+'''
+iLQR (batch formulation) applied to a Franka Emika manipulator for a viapoints task in a 3D workspace
+
+Copyright (c) 2024 Idiap Research Institute <https://www.idiap.ch/>
+Written by Philip Abbet <philip.abbet@idiap.ch>,
+Jérémy Maceiras <jeremy.maceiras@idiap.ch> and
+Sylvain Calinon <https://calinon.ch>
+
+This file is part of RCFS <https://robotics-codes-from-scratch.github.io/>
+License: MIT
+'''
+
+import numpy as np
+import numpy.matlib
+import matplotlib.pyplot as plt
+import matplotlib.patches as patches
+import math
+
+# Helper functions
+# ===============================
+
+# Cost and gradient for a viapoints reaching task (in object coordinate system)
+def f_reach(x, param):
+	ftmp, _ = fkin(x, param)
+	f = logmap(ftmp, param.Mu)
+	J = np.zeros([param.nbPoints * 6, param.nbPoints * param.nbVarX])
+	for t in range(param.nbPoints):
+		J[t*6:(t+1)*6, t*param.nbVarX:(t+1)*param.nbVarX] = Jkin_num(x[:,t], param)
+	return f, J
+
+# Forward kinematics for end-effector (from DH parameters)
+def fkin(x, param):
+	f = np.zeros((7, x.shape[1]))
+	R = np.zeros((3, 3, x.shape[1]))
+	for t in range(x.shape[1]):
+		ftmp, Rtmp = fkin0(x[:,t], param)
+		R[:,:,t] = Rtmp[:,:,-1]
+		f[0:3,t] = ftmp[:,-1]
+		f[3:7,t] = R2q(Rtmp[:,:,-1])
+	return f, R
+
+# Forward kinematics for all robot articulations, for articulatory joints and modified DH convention (a.k.a. Craig's formulation)
+def fkin0(x, param):
+	N = param.dh.q.shape[1]
+	x = np.append(x, 0)
+	Tf = np.eye(4)
+	R = np.zeros((3,3,N))
+	f = np.zeros((3,N+1))
+	for n in range(N):
+		ct = np.cos(x[n] + param.dh.q_offset[n])
+		st = np.sin(x[n] + param.dh.q_offset[n])
+		ca = np.cos(param.dh.alpha[n])
+		sa = np.sin(param.dh.alpha[n])
+		Tf = Tf @ [[ct,    -st,     0,   param.dh.r[n]   ],
+				   [st*ca,  ct*ca, -sa, -param.dh.d[n]*sa],
+				   [st*sa,  ct*sa,  ca,  param.dh.d[n]*ca],
+				   [0,      0,      0,   1               ]]
+		R[:,:,n] = Tf[0:3,0:3]
+		f[:,n+1] = Tf[0:3,-1]
+	return f, R
+
+# Jacobian of forward kinematics function with numerical computation
+def Jkin_num(x, param):
+	e = 1E-6
+	X = np.matlib.repmat(x, param.nbVarX, 1).T
+	F1, _ = fkin(X, param)
+	F2, _ = fkin(X + np.eye(param.nbVarX) * e, param)
+	J = logmap(F2, F1) / e # Error by considering manifold
+	return J
+
+# Logarithmic map for R^3 x S^3 manifold (with e in tangent space)
+def logmap(f, f0):
+	e = np.ndarray((6, f.shape[1]))
+	e[0:3, :] = f[0:3,:] - f0[0:3,:] # Error on R^3
+	for t in range(f.shape[1]):
+		H = dQuatToDxJac(f0[3:7,t])
+		e[3:6,t] = 2 * H @ logmap_S3(f[3:7,t], f0[3:7,t]) # Error on S^3
+	return e
+
+# Logarithmic map for S^3 manifold (with e in ambient space)
+def logmap_S3(x, x0):
+	th = acoslog(x0.T @ x)
+	if math.isnan(th):
+		th = 1.0
+
+	u = x - (x0.T @ x) * x0
+	u = np.multiply(th, u) / np.linalg.norm(u)
+
+	# import sys; sys.exit(0)
+	return u
+
+# Arcosine redefinition to make sure the distance between antipodal quaternions is zero
+def acoslog(x):
+	y = np.arccos(x)
+	if (x>=-1.0) and (x<0):
+		y = y - np.pi
+	return y
+
+def dQuatToDxJac(q):
+	return np.array([
+		[-q[1], q[0], -q[3], q[2]],
+		[-q[2], q[3], q[0], -q[1]],
+		[-q[3], -q[2], q[1], q[0]],
+	])
+
+# Unit quaternion to rotation matrix conversion (for quaternions as [w,x,y,z])
+def q2R(q):
+	return np.array([
+		[1.0 - 2.0 * q[2]**2 - 2.0 * q[3]**2, 2.0 * q[1] * q[2] - 2.0 * q[3] * q[0], 2.0 * q[1] * q[3] + 2.0 * q[2] * q[0]],
+		[2.0 * q[1] * q[2] + 2.0 * q[3] * q[0], 1.0 - 2.0 * q[1]**2 - 2.0 * q[3]**2, 2.0 * q[2] * q[3] - 2.0 * q[1] * q[0]],
+		[2.0 * q[1] * q[3] - 2.0 * q[2] * q[0], 2.0 * q[2] * q[3] + 2.0 * q[1] * q[0], 1.0 - 2.0 * q[1]**2 - 2.0 * q[2]**2],
+	])
+
+# Rotation matrix to unit quaternion conversion
+def R2q(R):
+	R = R.T
+	K = np.array([
+			[R[0,0]-R[1,1]-R[2,2], R[1,0]+R[0,1], R[2,0]+R[0,2], R[1,2]-R[2,1]],
+			[R[1,0]+R[0,1], R[1,1]-R[0,0]-R[2,2], R[2,1]+R[1,2], R[2,0]-R[0,2]],
+			[R[2,0]+R[0,2], R[2,1]+R[1,2], R[2,2]-R[0,0]-R[1,1], R[0,1]-R[1,0]],
+			[R[1,2]-R[2,1], R[2,0]-R[0,2], R[0,1]-R[1,0], R[0,0]+R[1,1]+R[2,2]],
+	]) / 3.0
+	_, V = np.linalg.eig(K)
+	return np.real([V[3, 0], V[0, 0], V[1, 0], V[2, 0]]) # for quaternions as [w,x,y,z]
+
+# Plot coordinate system
+def plotCoordSys(ax, x, R, width=1):
+	for t in range(x.shape[1]):
+		ax.plot([x[0,t], x[0,t]+R[0,0,t]], [x[1,t], x[1,t]+R[1,0,t]], [x[2,t], x[2,t]+R[2,0,t]], linewidth=2*width, color=[1.0, 0.0, 0.0])
+		ax.plot([x[0,t], x[0,t]+R[0,1,t]], [x[1,t], x[1,t]+R[1,1,t]], [x[2,t], x[2,t]+R[2,1,t]], linewidth=2*width, color=[0.0, 1.0, 0.0])
+		ax.plot([x[0,t], x[0,t]+R[0,2,t]], [x[1,t], x[1,t]+R[1,2,t]], [x[2,t], x[2,t]+R[2,2,t]], linewidth=2*width, color=[0.0, 0.0, 1.0])
+		ax.plot(x[0,t], x[1,t], x[2,t], 'o', markersize=10, color='black')
+
+
+## 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.nbIter = 30 # Maximum number of iterations for iLQR
+param.nbPoints = 2 # Number of viapoints
+param.nbVarX = 7 # State space dimension (x1,x2,x3)
+param.nbVarU = param.nbVarX # Control space dimension (dx1,dx2,dx3)
+param.nbVarF = 7 # Task space dimension (f1,f2,f3 for position, f4,f5,f6,f7 for unit quaternion)
+param.q = 1e0 # Tracking weighting term
+param.r = 1e-6 # Control weighting term
+
+Rtmp = q2R([np.cos(np.pi/3), np.sin(np.pi/3), 0.0, 0.0])
+param.MuR = np.dstack((Rtmp, Rtmp))
+param.Mu = np.ndarray((param.nbVarF, param.nbPoints))
+param.Mu[0:3, 0] = [.6, 0, .2]
+param.Mu[3:7, 0] = R2q(param.MuR[:,:,0])
+param.Mu[0:3, 1] = [.3, .5, .1]
+param.Mu[3:7, 1] = R2q(param.MuR[:,:,1])
+
+# DH parameters of Franka Emika robot
+param.dh = lambda: None # Lazy way to define an empty class in python
+param.dh.convention = 'm' # modified DH, a.k.a. Craig's formulation
+param.dh.type = ['r'] * (param.nbVarX+1) # Articulatory joints
+param.dh.q = np.zeros((1, param.nbVarX+1)) # Angle about previous z
+param.dh.q_offset = np.zeros((param.nbVarX+1,)) # Offset on articulatory joints
+param.dh.alpha = [0, -np.pi/2, np.pi/2, np.pi/2, -np.pi/2, np.pi/2, np.pi/2, 0] # Angle about common normal
+param.dh.d = [0.333, 0, 0.316, 0, 0.384, 0, 0, 0.107] # Offset along previous z to the common normal
+param.dh.r = [0, 0, 0, 0.0825, -0.0825, 0, 0.088, 0] # Length of the common normal
+
+
+# Main program
+# ===============================
+
+# Precision matrix
+Q = np.eye((param.nbVarF-1) * param.nbPoints) * param.q
+
+# Control weight matrix (at trajectory level)
+R = np.eye((param.nbData-1) * param.nbVarU) * param.r
+
+# 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)]).flatten()
+
+# 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.eye(param.nbVarX)).T
+Su = Su0[idx,:] # We remove the lines that are out of interest
+
+
+# iLQR
+# ===============================
+
+u = np.zeros(param.nbVarU * (param.nbData-1)) # Initial control command
+x0 = np.array([0, 0, 0, -np.pi/2, -0, np.pi/2, 0]) # Initial robot pose
+
+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_reach(x[:,tl], param) # Residuals and Jacobians
+	du = np.linalg.inv(Su.T @ J.T @ Q @ J @ Su + R) @ (-Su.T @ J.T @ Q @ f.flatten('F') - u * param.r) # Gauss-Newton update
+	# Estimate step size with backtracking line search method
+	alpha = 1
+	cost0 = f.flatten('F').T @ Q @ 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_reach(xtmp[:,tl], param) # Residuals 
+		cost = ftmp.flatten('F').T @ Q @ 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
+# ===============================
+
+ax = plt.figure(figsize=(12, 10)).add_subplot(projection='3d')
+
+# Plot the robot
+ftmp, _ = fkin0(x[:,0], param)
+ax.plot(ftmp[0,:], ftmp[1,:], ftmp[2,:], linewidth=4, color=[.8, .8, .8])
+
+ftmp, _ = fkin0(x[:,tl[0]], param)
+ax.plot(ftmp[0,:], ftmp[1,:], ftmp[2,:], linewidth=4, color=[.6, .6, .6])
+
+ftmp, _ = fkin0(x[:,tl[1]], param)
+ax.plot(ftmp[0,:], ftmp[1,:], ftmp[2,:], linewidth=4, color=[.4, .4, .4])
+
+# Plot targets
+plotCoordSys(ax, param.Mu, param.MuR * 0.1)
+
+# Plot end-effector and trajectory
+ftmp, Rtmp = fkin(x, param)
+ax.plot(ftmp[0,:], ftmp[1,:], ftmp[2,:], linewidth=1, color='black')
+plotCoordSys(ax, ftmp[:,0:1], Rtmp[:,:,0:1] * .05, width=2)
+plotCoordSys(ax, ftmp[:,tl], Rtmp[:,:,tl] * .05, width=2)
+
+plotCoordSys(ax, np.zeros((3,1)), np.eye(3).reshape((3, 3, 1)) * .1);
+
+# Set axes limits and labels
+ax.set_xlim(0, 0.8)
+ax.set_ylim(0, 0.8)
+ax.set_zlim(0, 0.8)
+ax.set_xlabel(r'$f_1$')
+ax.set_ylabel(r'$f_2$')
+ax.set_zlabel(r'$f_3$')
+
+limits = np.array([getattr(ax, f'get_{axis}lim')() for axis in 'xyz'])
+ax.set_box_aspect(np.ptp(limits, axis=1))
+
+plt.show()