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Commit 6d7ee83a authored by Hakan GIRGIN's avatar Hakan GIRGIN
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Script updated according to the template in iLQR_manipulator.py

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python/iLQR_manipulator_CP.py
+ 101
− 110
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......@@ -31,64 +31,62 @@ from math import factorial
# ===============================
# Logarithmic map for R^2 x S^1 manifold
def logmap(f,f0):
position_error = f[:,:2] - f0[:,:2]
orientation_error = np.imag(np.log( np.exp( f0[:,-1]*1j ).conj().T * np.exp(f[:,-1]*1j).T )).conj().reshape((-1,1))
error = np.hstack(( position_error , orientation_error ))
return error
# Forward kinematics for E-E
def fkin(param,x):
x = x.T
A = np.tril(np.ones([param.nbVarX,param.nbVarX]))
f = np.vstack((param.linkLengths @ np.cos(A @ x),
param.linkLengths @ np.sin(A @ x),
np.mod(np.sum(x,0)+np.pi, 2*np.pi) - np.pi)) #x1,x2,o (orientation as single Euler angle for planar robot)
return f.T
# Forward Kinematics for all joints
def fkin0(param,x):
T = np.tril(np.ones([param.nbVarX,param.nbVarX]))
T2 = np.tril(np.matlib.repmat(param.linkLengths,len(x),1))
f = np.vstack((
T2 @ np.cos(T@x),
T2 @ np.sin(T@x)
)).T
f = np.vstack((
np.zeros(2),
f
))
def logmap(f, f0):
position_error = f[:2,:] - f0[:2,:]
orientation_error = np.imag(np.log(np.exp(f0[-1,:]*1j).conj().T * np.exp(f[-1,:]*1j).T)).conj()
diff = np.vstack([position_error, orientation_error])
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.vstack([
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(param,x):
T = np.tril( np.ones((len(x),len(x))) )
J = np.vstack((
-np.sin(T@x).T @ np.diag(param.linkLengths) @ T,
np.cos(T@x).T @ np.diag(param.linkLengths) @ T,
np.ones(len(x))
))
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
# Residual and Jacobian
def f_reach(param,x):
f = logmap(fkin(param,x),param.mu)
J = np.zeros(( len(x) * param.nbVarF , len(x) * param.nbVarX ))
for t in range(x.shape[0]):
f[t,:2] = param.A[t].T @ f[t,:2] # Object oriented fk
Jtmp = jkin(param,x[t])
Jtmp[:2] = param.A[t].T @ Jtmp[:2] # Object centered jacobian
# Residual and Jacobian for a viapoints reaching task (in object coordinate system)
def f_reach(x, param):
f = logmap(fkin(x, param), param.Mu)
J = np.zeros([param.nbPoints * param.nbVarF, param.nbPoints * param.nbVarX])
for t in range(param.nbPoints):
f[:2, t] = param.A[:, :, t].T @ f[:2, t] # Object oriented residual
Jtmp = Jkin(x[:, t], param)
Jtmp[:2] = param.A[:, :, t].T @ Jtmp[:2] # Object centered Jacobian
if param.useBoundingBox:
for i in range(2):
if abs(f[t,i]) < param.sizeObj[i]:
f[t,i] = 0
Jtmp[i]=0
if abs(f[i, t]) < param.sz[i]:
f[i, t] = 0
Jtmp[i] = 0
else:
f[t,i] -= np.sign(f[t,i]) * param.sizeObj[i]
J[ t*param.nbVarF:(t+1)*param.nbVarF , t*param.nbVarX:(t+1)*param.nbVarX] = Jtmp
return f,J
f[i, t] -= np.sign(f[i, t]) * param.sz[i]
J[t * param.nbVarF:(t + 1) * param.nbVarF, t * param.nbVarX:(t + 1) * param.nbVarX] = Jtmp
return f, J
# Building piecewise constant basis functions
def build_phi_piecewise(nb_data, nb_fct):
......@@ -131,37 +129,31 @@ param.nbPoints = 2 # Number of viapoints
param.nbVarX = 3 # State space dimension (x1,x2,x3)
param.nbVarU = 3 # Control space dimension (dx1,dx2,dx3)
param.nbVarF = 3 # Objective function dimension (f1,f2,f3, with f3 as orientation)
param.linkLengths = [2,2,1] # Robot links lengths
param.sizeObj = [.2,.3] # Size of objects
param.l = [2,2,1] # Robot links lengths
param.sz = [.2,.3] # Size of objects
param.r = 1e-6 # Control weight term
param.mu = np.asarray([[2,1,-np.pi/6], [3,2,-np.pi/3]]).T # Viapoints
param.Mu = np.asarray([[2,1,-np.pi/2], [3,1,-np.pi/2]]).T # Viapoints
param.useBoundingBox = True # Consider bounding boxes for reaching cost
param.nbFct = 5 # Number of basis functions
param.A = np.zeros([2, 2, param.nbPoints]) # Object orientation matrices
param.basisName = "PIECEWISE"
# Task parameters
# Main program
# ===============================
# Targets
param.mu = np.asarray([
[2,1,-np.pi/2], # x , y , orientation
[3,1,-np.pi/2]
])
# Transformation matrices
param.A = np.zeros( (param.nbPoints,2,2) )
for i in range(param.nbPoints):
orn_t = param.mu[i,-1]
param.A[i,:,:] = np.asarray([
[np.cos(orn_t) , -np.sin(orn_t)],
[np.sin(orn_t) , np.cos(orn_t)]
# Object rotation matrices
for t in range(param.nbPoints):
orn_t = param.Mu[-1,t]
param.A[:,:,t] = np.asarray([
[np.cos(orn_t), -np.sin(orn_t)],
[np.sin(orn_t), np.cos(orn_t)]
])
# Regularization matrix
R = np.identity( (param.nbData-1) * param.nbVarU ) * param.r
# Precision matrix
Q = np.identity( param.nbVarF * param.nbPoints)
Q = np.identity(param.nbVarF * param.nbPoints)
# Control weight matrix
R = np.identity((param.nbData-1) * param.nbVarU) * param.r
# System parameters
# ===============================
......@@ -171,13 +163,13 @@ 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)])
u = np.zeros( param.nbVarU * (param.nbData-1) ) # Initial control command
x0 = np.array( [3*np.pi/4 , -np.pi/2 , - np.pi/4] ) # Initial state (in joint space)
# 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
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
# Basis functions
......@@ -195,33 +187,32 @@ PSI = np.kron(phi,np.identity(param.nbVarU))
# Solving iLQR
# ===============================
for i in range( param.nbIter ):
x = np.real(Su0 @ u + Sx0 @ x0)
x = x.reshape( ( param.nbData , param.nbVarX) )
u = np.zeros( param.nbVarU * (param.nbData-1) ) # Initial control command
x0 = np.array( [3*np.pi/4 , -np.pi/2 , - np.pi/4] ) # Initial state (in joint space)
f, J = f_reach(param,x[tl])
dw = np.linalg.inv(PSI.T @ Su.T @ J.T @ Q @ J @ Su @ PSI + PSI.T @ R @ PSI) @ (-PSI.T @ Su.T @ J.T @ Q @ f.flatten() - PSI.T @ u * param.r)
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
dw = np.linalg.inv(PSI.T @ Su.T @ J.T @ Q @ J @ Su @ PSI + PSI.T @ R @ PSI) @\
(-PSI.T @ Su.T @ J.T @ Q @ f.flatten('F') - PSI.T @ u * param.r)
du = PSI @ dw
# Perform line search
# Estimate step size with backtracking line search method
alpha = 1
cost0 = f.flatten() @ Q @ f.flatten() + np.linalg.norm(u) * param.r
cost0 = f.flatten('F').T @ Q @ f.flatten('F') + np.linalg.norm(u)**2 * param.r # Cost
while True:
utmp = u + du * alpha
xtmp = np.real(Su0 @ utmp + Sx0 @ x0)
xtmp = xtmp.reshape( ( param.nbData , param.nbVarX) )
ftmp, _ = f_reach(param,xtmp[tl])
cost = ftmp.flatten() @ Q @ ftmp.flatten() + np.linalg.norm(utmp) * param.r
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: {}, alpha: {}".format(i,cost,alpha))
print("Iteration {}, cost: {}".format(i,cost))
break
alpha /=2
if np.linalg.norm( alpha * du ) < 1e-2:
break
if np.linalg.norm(du * alpha) < 1E-2:
break # Stop iLQR iterations when solution is reached
# Plotting
# ===============================
......@@ -231,17 +222,17 @@ plt.axis("off")
plt.gca().set_aspect('equal', adjustable='box')
# Get points of interest
f = fkin(param,x)
f00 = fkin0(param,x[0])
f10 = fkin0(param,x[tl[0]])
fT0 = fkin0(param,x[-1])
f = fkin(x, param)
f00 = fkin0(x[:, 0], param)
f10 = fkin0(x[:, tl[0]], param)
fT0 = fkin0(x[:, -1], param)
u = u.reshape((-1,param.nbVarU))
plt.plot( f00[:,0] , f00[:,1],c='black',linewidth=5,alpha=.2)
plt.plot( f10[:,0] , f10[:,1],c='black',linewidth=5,alpha=.4)
plt.plot( fT0[:,0] , fT0[:,1],c='black',linewidth=5,alpha=.6)
plt.plot( f00[0, :] , f00[1,:],c='black',linewidth=5,alpha=.2)
plt.plot( f10[0, :] , f10[1,:],c='black',linewidth=5,alpha=.4)
plt.plot( fT0[0, :] , fT0[1,:],c='black',linewidth=5,alpha=.6)
plt.plot(f[:,0],f[:,1],c="black",marker="o",markevery=[0]+tl.tolist()) #,label="Trajectory"2
plt.plot(f[0, :],f[1, :],c="black",marker="o",markevery=[0]+tl.tolist()) #,label="Trajectory"2
# Plot bounding box or via-points
ax = plt.gca()
......@@ -249,36 +240,36 @@ color_map = ["deepskyblue","darkorange"]
for i in range(param.nbPoints):
if param.useBoundingBox:
rect_origin = param.mu[i,:2] - param.A[i]@np.array(param.sizeObj)
rect_orn = param.mu[i,-1]
rect_origin = param.Mu[i,:2] - param.A[i]@np.array(param.sz)
rect_orn = param.Mu[i,-1]
rect = patches.Rectangle(rect_origin,param.sizeObj[0]*2,param.sizeObj[1]*2,np.degrees(rect_orn),color=color_map[i])
rect = patches.Rectangle(rect_origin,param.sz[0]*2,param.sz[1]*2,np.degrees(rect_orn),color=color_map[i])
ax.add_patch(rect)
else:
plt.scatter(param.mu[i,0],param.mu[i,1],s=100,marker="X",c=color_map[i])
plt.scatter(param.Mu[i,0],param.Mu[i,1],s=100,marker="X",c=color_map[i])
fig,axs = plt.subplots(7,1)
axs[0].plot(f[:,0],c='black')
axs[0].plot(f[0],c='black')
axs[0].set_ylabel("$f(x)_1$")
axs[0].set_xticks([0,param.nbData])
axs[0].set_xticklabels(["0","T"])
for i in range(param.nbPoints):
axs[0].scatter(tl[i],param.mu[i,0],c='blue')
axs[0].scatter(tl[i],param.Mu[0, i],c='blue')
axs[1].plot(f[:,1],c='black')
axs[1].plot(f[1],c='black')
axs[1].set_ylabel("$f(x)_2$")
axs[1].set_xticks([0,param.nbData])
axs[1].set_xticklabels(["0","T"])
for i in range(param.nbPoints):
axs[1].scatter(tl[i],param.mu[i,1],c='blue')
axs[1].scatter(tl[i],param.Mu[1, i],c='blue')
axs[2].plot(f[:,2],c='black')
axs[2].plot(f[2],c='black')
axs[2].set_ylabel("$f(x)_3$")
axs[2].set_xticks([0,param.nbData])
axs[2].set_xticklabels(["0","T"])
for i in range(param.nbPoints):
axs[2].scatter(tl[i],param.mu[i,2],c='blue')
axs[2].scatter(tl[i],param.Mu[2, i],c='blue')
axs[3].plot(u[:,0],c='black')
axs[3].set_ylabel("$u_1$")
......
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