def ComputeTargetPosture(self,
x_seed,
target_manipuland_pose,
backoff_distance=0.0):
q_des_full = np.zeros(self.nq)
q_des_full[self.nq - 3:] = target_manipuland_pose
desired_positions = self.transform_grasp_points_manipuland(q_des_full)
desired_normals = self.transform_grasp_normals_manipuland(q_des_full)
desired_positions -= backoff_distance * desired_normals # back off a bit
desired_angles = [
math.atan2(desired_normals[1, k], desired_normals[0, k])
for k in range(desired_normals.shape[1])
]
constraints = []
# Constrain the fingertips to lie on the grasp points,
# facing approximately along the grasp angles.
for i in range(len(self.grasp_points)):
constraints.append(
ik.WorldPositionConstraint(
self.hand,
self.finger_link_indices[self.grasp_finger_assignments[i]
[0]],
self.grasp_finger_assignments[i][1],
desired_positions[:, i] - 0.01, # lb
desired_positions[:, i] + 0.01) # ub
)
constraints.append(
ik.WorldEulerConstraint(
self.hand,
self.finger_link_indices[self.grasp_finger_assignments[i]
[0]],
[-0.01, -0.01, desired_angles[i] - 1.0], # lb
[0.01, 0.01, desired_angles[i] + 1.0]) # ub
)
posture_constraint = ik.PostureConstraint(self.hand)
# Disambiguate the continuous rotation joint angle choices
# for the hand.
posture_constraint.setJointLimits(np.arange(self.nq - 3),
-np.ones(self.nq - 3) * math.pi,
np.ones(self.nq - 3) * math.pi)
# Constrain the manipuland to be in the target position.
posture_constraint.setJointLimits(
[self.nq - 3, self.nq - 2, self.nq - 1],
target_manipuland_pose - 0.01, target_manipuland_pose + 0.01)
constraints.append(posture_constraint)
options = ik.IKoptions(self.hand)
results = ik.InverseKin(self.hand, x_seed[0:self.nq],
self.x_nom[0:self.nq], constraints, options)
return results.q_sol[0], results.info[0]
def handlePostureConstraint(self, c, fields):
tspan = np.asarray(c.tspan, dtype=float)
pose = np.asarray(fields.poses[c.postureName], dtype=float)
positionInds = np.array([self.positionNameToId[name] for name in c.joints], dtype=np.int32)
lowerLimit = pose[positionInds] + c.jointsLowerBound
upperLimit = pose[positionInds] + c.jointsUpperBound
pc = pydrakeik.PostureConstraint(self.rigidBodyTree, tspan)
pc.setJointLimits(positionInds, lowerLimit, upperLimit)
return pc
def plan_ee_configuration(rbt,
q0,
qseed,
ee_pose,
ee_body,
ee_point,
allow_collision=True,
active_bodies_idx=list(),
euler_limits=0.3):
''' Performs IK for a single point in time
to get the Kuka's gripper to a specified
pose in space. '''
nq = rbt.get_num_positions()
controlled_joint_names = [
"iiwa_joint_1", "iiwa_joint_2", "iiwa_joint_3", "iiwa_joint_4",
"iiwa_joint_5", "iiwa_joint_6", "iiwa_joint_7"
]
free_config_inds, constrained_config_inds = \
kuka_utils.extract_position_indices(rbt, controlled_joint_names)
# Assemble IK constraints
constraints = []
if not allow_collision:
constraints.append(
ik.MinDistanceConstraint(model=rbt,
min_distance=1E-6,
active_bodies_idx=active_bodies_idx,
active_group_names=set()))
# Constrain the non-searched-over joints
posture_constraint = ik.PostureConstraint(rbt)
posture_constraint.setJointLimits(constrained_config_inds,
q0[constrained_config_inds] - 0.001,
q0[constrained_config_inds] + 0.001)
constraints.append(posture_constraint)
# Constrain the ee frame to lie on the target point
# facing in the target orientation
constraints.append(
ik.WorldPositionConstraint(rbt, ee_body, ee_point, ee_pose[0:3] - 0.01,
ee_pose[0:3] + 0.01))
constraints.append(
ik.WorldEulerConstraint(rbt, ee_body, ee_pose[3:6] - euler_limits,
ee_pose[3:6] + euler_limits))
options = ik.IKoptions(rbt)
results = ik.InverseKin(rbt, q0, qseed, constraints, options)
#print "plan ee config info %d" % results.info[0]
return results.q_sol[0], results.info[0]
def testPostureConstraint(self):
r = pydrake.rbtree.RigidBodyTree(
os.path.join(pydrake.getDrakePath(),
"examples/Pendulum/Pendulum.urdf"))
q = -0.9
posture_constraint = ik.PostureConstraint(r)
posture_constraint.setJointLimits(np.array([[6]], dtype=np.int32),
np.array([[q]]),
np.array([[q]]))
# Choose a seed configuration (randomly) and a nominal configuration
# (at 0)
q_seed = np.vstack((np.zeros((6, 1)),
0.8147))
q_nom = np.vstack((np.zeros((6, 1)),
0.))
options = ik.IKoptions(r)
results = ik.InverseKin(
r, q_seed, q_nom, [posture_constraint], options)
self.assertEqual(results.info[0], 1)
self.assertAlmostEqual(results.q_sol[0][6], q, 1e-9)
# Run the tests again both pointwise and as a trajectory to
# validate the interfaces.
t = np.array([0., 1.])
q_seed_array = np.transpose(np.array([q_seed, q_seed]))[0]
q_nom_array = np.transpose(np.array([q_nom, q_nom]))[0]
results = ik.InverseKinPointwise(r, t, q_seed_array, q_nom_array,
[posture_constraint], options)
self.assertEqual(len(results.info), 2)
self.assertEqual(len(results.q_sol), 2)
self.assertEqual(results.info[0], 1)
# The pointwise result will go directly to the constrained
# value.
self.assertAlmostEqual(results.q_sol[0][6], q, 1e-9)
self.assertEqual(results.info[1], 1)
self.assertAlmostEqual(results.q_sol[1][6], q, 1e-9)
results = ik.InverseKinTraj(r, t, q_seed_array, q_nom_array,
[posture_constraint], options)
self.assertEqual(len(results.info), 1)
self.assertEqual(len(results.q_sol), 2)
self.assertEqual(results.info[0], 1)
# The trajectory result starts at the initial value and moves
# to the constrained value.
self.assertAlmostEqual(results.q_sol[0][6], q_seed[6], 1e-9)
self.assertAlmostEqual(results.q_sol[1][6], q, 1e-9)
def testPostureConstraint(self):
r = pydrake.rbtree.RigidBodyTree(
os.path.join(pydrake.getDrakePath(),
"examples/Pendulum/Pendulum.urdf"))
q = -0.9
posture_constraint = ik.PostureConstraint(r)
posture_constraint.setJointLimits(np.array([[6]], dtype=np.int32),
np.array([[q]]), np.array([[q]]))
# Choose a seed configuration (randomly) and a nominal configuration (at 0)
q_seed = np.vstack((np.zeros((6, 1)), 0.8147))
q_nom = np.vstack((np.zeros((6, 1)), 0.))
options = ik.IKoptions(r)
results = ik.inverseKinSimple(r, q_seed, q_nom, [posture_constraint],
options)
self.assertAlmostEqual(results.q_sol[6], q, 1e-9)
def plan_grasping_configuration(rbt, q0, target_ee_pose):
''' Performs IK for a single point in time
to get the Kuka's gripper to a specified
pose in space. '''
nq = rbt.get_num_positions()
q_des_full = np.zeros(nq)
controlled_joint_names = [
"iiwa_joint_1", "iiwa_joint_2", "iiwa_joint_3", "iiwa_joint_4",
"iiwa_joint_5", "iiwa_joint_6", "iiwa_joint_7"
]
free_config_inds, constrained_config_inds = \
kuka_utils.extract_position_indices(rbt, controlled_joint_names)
# Assemble IK constraints
constraints = []
# Constrain the non-searched-over joints
posture_constraint = ik.PostureConstraint(rbt)
posture_constraint.setJointLimits(constrained_config_inds,
q0[constrained_config_inds] - 0.01,
q0[constrained_config_inds] + 0.01)
constraints.append(posture_constraint)
# Constrain the ee frame to lie on the target point
# facing in the target orientation
ee_frame = rbt.findFrame("iiwa_frame_ee").get_frame_index()
constraints.append(
ik.WorldPositionConstraint(rbt, ee_frame, np.zeros(
(3, 1)), target_ee_pose[0:3] - 0.01, target_ee_pose[0:3] + 0.01))
constraints.append(
ik.WorldEulerConstraint(rbt, ee_frame, target_ee_pose[3:6] - 0.01,
target_ee_pose[3:6] + 0.01))
options = ik.IKoptions(rbt)
results = ik.InverseKin(rbt, q0, q0, constraints, options)
print results.q_sol, "info %d" % results.info[0]
return results.q_sol[0], results.info[0]
def plan_grasping_trajectory(rbt, q0, target_reach_pose, target_grasp_pose,
n_knots, reach_time, grasp_time):
''' Solves IK at a series of sample times (connected with a
cubic spline) to generate a trajectory to bring the Kuka from an
initial pose q0 to a final end effector pose in the specified
time, using the specified number of knot points.
Uses an intermediate pose reach_pose as an intermediate target
to hit at the knot point closest to reach_time.
See http://drake.mit.edu/doxygen_cxx/rigid__body__ik_8h.html
for the "inverseKinTraj" entry. At the moment, the Python binding
for this function uses "inverseKinTrajSimple" -- i.e., it doesn't
return derivatives. '''
nq = rbt.get_num_positions()
q_des_full = np.zeros(nq)
# Create knot points
ts = np.linspace(0., grasp_time, n_knots)
# Figure out the knot just before reach time
reach_start_index = np.argmax(ts >= reach_time) - 1
controlled_joint_names = [
"iiwa_joint_1", "iiwa_joint_2", "iiwa_joint_3", "iiwa_joint_4",
"iiwa_joint_5", "iiwa_joint_6", "iiwa_joint_7"
]
free_config_inds, constrained_config_inds = \
kuka_utils.extract_position_indices(rbt, controlled_joint_names)
# Assemble IK constraints
constraints = []
# Constrain the non-searched-over joints for all time
all_tspan = np.array([0., grasp_time])
posture_constraint = ik.PostureConstraint(rbt, all_tspan)
posture_constraint.setJointLimits(constrained_config_inds,
q0[constrained_config_inds] - 0.01,
q0[constrained_config_inds] + 0.01)
constraints.append(posture_constraint)
# Constrain all joints to be the initial posture at the start time
start_tspan = np.array([0., 0.])
posture_constraint = ik.PostureConstraint(rbt, start_tspan)
posture_constraint.setJointLimits(free_config_inds,
q0[free_config_inds] - 0.01,
q0[free_config_inds] + 0.01)
constraints.append(posture_constraint)
# Constrain the ee frame to lie on the target point
# facing in the target orientation in between the
# reach and final times
ee_frame = rbt.findFrame("iiwa_frame_ee").get_frame_index()
for i in range(reach_start_index, n_knots):
this_tspan = np.array([ts[i], ts[i]])
interp = float(i - reach_start_index) / (n_knots - reach_start_index)
target_pose = (1. -
interp) * target_reach_pose + interp * target_grasp_pose
constraints.append(
ik.WorldPositionConstraint(rbt,
ee_frame,
np.zeros((3, 1)),
target_pose[0:3] - 0.01,
target_pose[0:3] + 0.01,
tspan=this_tspan))
constraints.append(
ik.WorldEulerConstraint(rbt,
ee_frame,
target_pose[3:6] - 0.05,
target_pose[3:6] + 0.05,
tspan=this_tspan))
# Seed and nom are both the initial repeated for the #
# of knot points
q_seed = np.tile(q0, [1, n_knots])
q_nom = np.tile(q0, [1, n_knots])
options = ik.IKoptions(rbt)
# Set bounds on initial and final velocities
zero_velocity = np.zeros(rbt.get_num_velocities())
options.setqd0(zero_velocity, zero_velocity)
options.setqdf(zero_velocity, zero_velocity)
results = ik.InverseKinTraj(rbt, ts, q_seed, q_nom, constraints, options)
qtraj = PiecewisePolynomial.Pchip(ts, np.vstack(results.q_sol).T, True)
print "IK returned a solution with info %d" % results.info[0]
print "(Info 1 is good, other values are dangerous)"
return qtraj, results.info[0]
def trajectoryIK(self, start, end):
dur = self.config["trajectory_duration"]
joints = np.array(
[self.positions[joint] for joint in self.joint_names],
dtype=np.int32
).reshape(9, 1)
constraints = list()
# Constrain the start position
start_pose = np.concatenate(start).reshape(9, 1)
start_constraint = ik.PostureConstraint( self.robot,
np.array([0.0, 0.0]).reshape([2,1]) # Active time
)
start_constraint.setJointLimits(
joints,
start_pose - self.config['pose_tol'],
start_pose + self.config['pose_tol']
)
constraints.append(start_constraint)
# Constrain the end position
end_pose = np.concatenate(end).reshape(9, 1)
end_constraint = ik.PostureConstraint( self.robot,
np.array([dur, dur]).reshape([2,1]) # Active time
)
end_constraint.setJointLimits(
joints,
end_pose - self.config['pose_tol'],
end_pose + self.config['pose_tol']
)
constraints.append(end_constraint)
# Constrain against collisions
constraints.append( ik.MinDistanceConstraint ( self.robot,
self.config['collision_min_distance'], # Minimum distance between bodies
list(), # Active bodies (empty set means all bodies)
set() # Active collision groups (not filter groups! Empty set means all)
))
# Prepare times of interest for trajectory solve
times = np.linspace(0, dur, num=self.config['trajectory_count'], endpoint=True)
# Compute cubic interpolation as seed trajectory
x = np.array([0, dur])
y = np.hstack([start_pose, end_pose])
f = CubicSpline(x, y, axis=1, bc_type='clamped')
q_seed = f(times)
# print "q_seed:", q_seed
# Solve the IK problem
options = ik.IKoptions(self.robot)
options.setMajorIterationsLimit(400)
options.setFixInitialState(True)
# print "Iterations limit:", options.getMajorIterationsLimit()
result = ik.InverseKinTraj(self.robot, times, q_seed, q_seed, constraints, options)
if result.info[0] != 1:
print "Could not find safe trajectory!"
print "Start pose:", start_pose.flatten()
print "End pose: ", end_pose.flatten()
print "Result:", result.info
return None
else:
# Clean up: Pack solutions as a matrix, with time as first column
q_sol = np.vstack(result.q_sol)
q_sol = np.insert(q_sol, [0], times[:, None], axis=1)
return q_sol
def computePoses(self):
# Precompute poses
head_poses = self.computeHeadPoses()
plate_poses = self.computePlatePoses()
print "Total head poses:", len(head_poses)
print "Total plate poses:", len(plate_poses)
print
if self.config["save_ik"]:
filename = self.config["ik_save_path"]
print "IK save path: {}".format(filename)
if os.path.isfile(filename):
self.config["save_ik"] = False
print "WARNING! IK save path already exists. IK saving disabled"
print
print "Performing IK..."
# Solve all the IK
all_poses = list()
for i, head_pose in enumerate(head_poses):
for plate_pose in plate_poses:
# Generate constraints and solve IK
constraints = list()
# Constrain the head pose
head_constraint = ik.PostureConstraint( self.robot )
head_constraint.setJointLimits(
np.array([self.positions['ptu_pan'], self.positions['ptu_tilt']], dtype=np.int32).reshape(2, 1),
head_pose - self.config['head_pose_tol'],
head_pose + self.config['head_pose_tol']
)
constraints.append(head_constraint)
# Set the calibration plate position
constraints.append( ik.RelativePositionConstraint ( self.robot,
np.zeros([3,1]), # Point relative to Body A (calibration target)
plate_pose - self.config['arm_pos_tol'], # Lower bound relative to Body B (optical frame)
plate_pose + self.config['arm_pos_tol'], # Upper bound relative to Body B (optical frame)
self.bodies['calibration_target'], # Body A
self.bodies[self.config['optical_frame']], # Body B
np.concatenate([ # Transform from B to reference point
np.zeros(3), # Translation (identity)
np.array([1, 0, 0, 0]) # Rotation (identity)
]).reshape(7, 1)
))
# Set the calibration plate orientation
constraints.append( ik.RelativeGazeDirConstraint ( self.robot,
self.bodies['calibration_target'], # Body A
self.bodies[self.config['optical_frame']], # Body B
np.array([0, 0, 1], dtype=np.float64).reshape(3,1), # Body A axis
np.array([0, 0, -1], dtype=np.float64).reshape(3,1), # Body B axis
self.config['gaze_dir_tol'] # Cone tolerance in radians
))
# Avoid self-collision
constraints.append( ik.MinDistanceConstraint ( self.robot,
self.config['collision_min_distance'], # Minimum distance between bodies
list(), # Active bodies (empty set means all bodies)
set() # Active collision groups (not filter groups! Empty set means all)
))
# Actually solve the IK!
# Since we're solving ahead of time, just use the zero as seed
# NOTE: Could possibly track last solution as seed...
q_seed = self.robot.getZeroConfiguration()
options = ik.IKoptions(self.robot)
result = ik.InverseKin(self.robot, q_seed, q_seed, constraints, options)
if result.info[0] != 1:
pass
else:
# Tuple: head_pose, arm_pose
all_poses.append((result.q_sol[0][0:2],result.q_sol[0][2:]))
if self.config["save_ik"]:
with open(self.config["ik_save_path"],'a') as fp:
np.savetxt(fp, result.q_sol[0][None, :], delimiter=',')
# Show status info after every head pose
attempted = (i+1) * len(plate_poses)
success = len(all_poses)
total = len(head_poses)*len(plate_poses)
completion = attempted*100 / total
print "[{:>3d}%] {}/{} solutions found".format(completion, success, attempted)
return all_poses
def plan_trajectory_to_posture(rbt, q0, qf, n_knots, duration, start_time=0.):
''' Solves IK at a series of sample times (connected with a
cubic spline) to generate a trajectory to bring the Kuka from an
initial pose q0 to a final end effector pose in the specified
time, using the specified number of knot points.
Uses an intermediate pose reach_pose as an intermediate target
to hit at the knot point closest to reach_time.
See http://drake.mit.edu/doxygen_cxx/rigid__body__ik_8h.html
for the "inverseKinTraj" entry. At the moment, the Python binding
for this function uses "inverseKinTrajSimple" -- i.e., it doesn't
return derivatives. '''
nq = rbt.get_num_positions()
# Create knot points
ts = np.linspace(0., duration, n_knots)
controlled_joint_names = [
"iiwa_joint_1", "iiwa_joint_2", "iiwa_joint_3", "iiwa_joint_4",
"iiwa_joint_5", "iiwa_joint_6", "iiwa_joint_7"
]
free_config_inds, constrained_config_inds = \
kuka_utils.extract_position_indices(rbt, controlled_joint_names)
# Assemble IK constraints
constraints = []
q0 = np.clip(q0, rbt.joint_limit_min, rbt.joint_limit_max)
qf = np.clip(qf, rbt.joint_limit_min[free_config_inds],
rbt.joint_limit_max[free_config_inds])
#constraints.append(ik.MinDistanceConstraint(
# model=rbt, min_distance=0.01, active_bodies_idx=list(),
# active_group_names=set()))
# Constrain the non-searched-over joints for all time
all_tspan = np.array([0., duration])
posture_constraint = ik.PostureConstraint(rbt, all_tspan)
posture_constraint.setJointLimits(constrained_config_inds,
q0[constrained_config_inds] - 0.01,
q0[constrained_config_inds] + 0.01)
constraints.append(posture_constraint)
# Constrain all joints to be the initial posture at the start time
# TODO: actually freeze these, rather than letting them be searched over.
# No point actually searching over these, and makes IK way slower
start_tspan = np.array([0., 0.])
posture_constraint = ik.PostureConstraint(rbt, start_tspan)
posture_constraint.setJointLimits(free_config_inds,
q0[free_config_inds] - 0.01,
q0[free_config_inds] + 0.01)
constraints.append(posture_constraint)
# Constrain constrainted joints to be the final posture at the final time
end_tspan = np.array([duration, duration])
posture_constraint = ik.PostureConstraint(rbt, end_tspan)
posture_constraint.setJointLimits(free_config_inds,
qf[free_config_inds] - 0.01,
qf[free_config_inds] + 0.01)
constraints.append(posture_constraint)
# Seed is joint-space linear interpolation of the
# initial and final posture (a reasonable initial guess)
q_seed = np.zeros([q0.shape[0], n_knots])
qf_full = q0 * 0.
qf_full[free_config_inds] = qf
for i, t in enumerate(ts):
frac = t / duration
q_seed[:, i] = q0 * (1. - frac) + qf_full * (frac)
# Nominal is the final posture
q_nom = np.tile(qf_full, [1, n_knots])
options = ik.IKoptions(rbt)
# Set bounds on initial and final velocities
zero_velocity = np.zeros(rbt.get_num_velocities())
options.setqd0(zero_velocity, zero_velocity)
options.setqdf(zero_velocity, zero_velocity)
results = ik.InverseKinTraj(rbt, ts, q_seed, q_nom, constraints, options)
ts += start_time
qtraj = PiecewisePolynomial.Pchip(ts, np.vstack(results.q_sol).T, True)
print "IK returned a solution with info %d" % results.info[0]
return qtraj, results.info[0]
def plan_ee_trajectory(rbt,
q0,
traj_seed,
ee_times,
ee_poses,
ee_body,
ee_point,
n_knots,
start_time=0.,
avoid_collision_tspans=[],
active_bodies_idx=list()):
nq = rbt.get_num_positions()
# Create knot points
ts = np.linspace(0., ee_times[-1], n_knots)
# Insert one more for each ee time if not already in there
for t in ee_times:
if t not in ts:
ts = np.insert(ts, np.argmax(ts >= t), t)
print "times ", ts, " for times ", ee_times
controlled_joint_names = [
"iiwa_joint_1", "iiwa_joint_2", "iiwa_joint_3", "iiwa_joint_4",
"iiwa_joint_5", "iiwa_joint_6", "iiwa_joint_7"
]
free_config_inds, constrained_config_inds = \
kuka_utils.extract_position_indices(rbt, controlled_joint_names)
# Assemble IK constraints
constraints = []
for tspan in avoid_collision_tspans:
constraints.append(
ik.MinDistanceConstraint(model=rbt,
min_distance=1E-6,
active_bodies_idx=active_bodies_idx,
active_group_names=set(),
tspan=tspan))
print "active for ", tspan
# Make starting constraint sensible if it's out of joint limits
q0 = np.clip(q0, rbt.joint_limit_min, rbt.joint_limit_max)
# Constrain the non-searched-over joints for all time
all_tspan = np.array([0., ts[-1]])
posture_constraint = ik.PostureConstraint(rbt, all_tspan)
posture_constraint.setJointLimits(constrained_config_inds,
q0[constrained_config_inds] - 0.05,
q0[constrained_config_inds] + 0.05)
constraints.append(posture_constraint)
# Constrain all joints to be the initial posture at the start time
start_tspan = np.array([0., 0.])
posture_constraint = ik.PostureConstraint(rbt, start_tspan)
posture_constraint.setJointLimits(free_config_inds,
q0[free_config_inds] - 0.01,
q0[free_config_inds] + 0.01)
constraints.append(posture_constraint)
# Constrain the ee frame to lie on the target point
# facing in the target orientation in between the
# reach and final times)
for t, pose in zip(ee_times, ee_poses):
this_tspan = np.array([t - 0.01, t + 0.01])
constraints.append(
ik.WorldPositionConstraint(rbt,
ee_body,
ee_point,
pose[0:3] - 0.01,
pose[0:3] + 0.01,
tspan=this_tspan))
constraints.append(
ik.WorldEulerConstraint(rbt,
ee_body,
pose[3:6] - 0.05,
pose[3:6] + 0.05,
tspan=this_tspan))
# Seed and nom are both the initial repeated for the #
# of knot points
q_seed = np.hstack([traj_seed.value(t) for t in ts])
q_nom = np.tile(q0, [ts.shape[0], 1]).T
options = ik.IKoptions(rbt)
# Set bounds on initial and final velocities
zero_velocity = np.zeros(rbt.get_num_velocities())
options.setqd0(zero_velocity, zero_velocity)
options.setqdf(zero_velocity, zero_velocity)
Q = np.eye(q0.shape[0], q0.shape[0])
Q[free_config_inds[-2:], free_config_inds[-2:]] *= 1
options.setQ(Q)
options.setQv(np.eye(q0.shape[0], q0.shape[0]) * 10)
results = ik.InverseKinTraj(rbt, ts, q_nom, q_seed, constraints, options)
qtraj = PiecewisePolynomial.Pchip(ts + start_time,
np.vstack(results.q_sol).T, True)
print "IK returned a solution with info %d" % results.info[0]
return qtraj, results.info[0], results.q_sol
