diff --git a/crocoddyl/__init__.py b/crocoddyl/__init__.py
index f128f612dffc99a6ca82388db41d6eab477dcb7f..ecca59a228e0eca8b602c8cb1b3442cc7442c218 100644
--- a/crocoddyl/__init__.py
+++ b/crocoddyl/__init__.py
@@ -3,7 +3,7 @@ from .action import (ActionDataAbstract, ActionDataLQR, ActionDataNumDiff, Actio
from .activation import (ActivationDataInequality, ActivationDataQuad, ActivationDataSmoothAbs,
ActivationDataWeightedQuad, ActivationModelInequality, ActivationModelQuad,
ActivationModelSmoothAbs, ActivationModelWeightedQuad)
-from .actuation import ActuationDataFreeFloating, ActuationDataFull, ActuationModelFreeFloating, ActuationModelFull
+from .actuation import ActuationDataFreeFloating, ActuationDataFull, ActuationModelFreeFloating, ActuationModelFull, ActuationModelUAM
from .box_ddp import SolverBoxDDP
from .box_kkt import SolverBoxKKT
from .callbacks import CallbackDDPLogger, CallbackDDPVerbose, CallbackSolverDisplay, CallbackSolverTimer
@@ -23,12 +23,13 @@ from .differential_action import (DifferentialActionDataAbstract, DifferentialAc
DifferentialActionModelLQR, DifferentialActionModelNumDiff)
from .fddp import SolverFDDP
from .floating_contact import DifferentialActionDataFloatingInContact, DifferentialActionModelFloatingInContact
+from .flying import DifferentialActionModelUAM, DifferentialActionDataUAM
from .impact import (ActionDataImpact, ActionModelImpact, CostModelImpactCoM, CostModelImpactWholeBody, ImpulseData6D,
ImpulseDataPinocchio, ImpulseModel3D, ImpulseModel6D, ImpulseModelMultiple, ImpulseModelPinocchio)
from .integrated_action import (IntegratedActionDataEuler, IntegratedActionDataRK4, IntegratedActionModelEuler,
IntegratedActionModelRK4)
from .kkt import SolverKKT
-from .robots import getTalosPathFromRos, loadHyQ, loadTalos, loadTalosArm, loadTalosLegs
+from .robots import getTalosPathFromRos, loadHyQ, loadTalos, loadTalosArm, loadTalosLegs, loadKinton, loadKintonArm
from .shooting import ShootingProblem
from .solver import SolverAbstract
from .state import StateAbstract, StateNumDiff, StatePinocchio, StateVector
diff --git a/crocoddyl/actuation.py b/crocoddyl/actuation.py
index 3e416ec2e7a4596043407d362a6ad85d20b7302f..dac17e43c6694b652c3e926aff87a55eab5e107d 100644
--- a/crocoddyl/actuation.py
+++ b/crocoddyl/actuation.py
@@ -2,6 +2,44 @@ import warnings
import numpy as np
+class ActuationModelUAM:
+ '''
+ This model transforms an actuation u into a joint torque tau.
+ We implement here the simplest model: tau = S.T*u, where S is constant.
+ '''
+
+ def __init__(self, pinocchioModel):
+ self.pinocchio = pinocchioModel
+ if (pinocchioModel.joints[1].shortname() != 'JointModelFreeFlyer'):
+ warnings.warn('Strange that the first joint is not a freeflyer')
+ self.nq = pinocchioModel.nq
+ self.nv = pinocchioModel.nv
+ self.nx = self.nq + self.nv
+ self.ndx = self.nv * 2
+ self.nu = self.nv - 2
+
+ def calc(self, data, x, u):
+ data.a[2:] = u
+ return data.a
+
+ def calcDiff(self, data, x, u, recalc=True):
+ if recalc:
+ self.calc(data, x, u)
+ return data.a
+
+ def createData(self, pinocchioData):
+ return ActuationDataUAM(self, pinocchioData)
+
+
+class ActuationDataUAM:
+ def __init__(self, model, pinocchioData):
+ self.pinocchio = pinocchioData
+ ndx, nv, nu = model.ndx, model.nv, model.nu
+ self.a = np.zeros(nv) # result of calc
+ self.A = np.zeros([nv, ndx + nu]) # result of calcDiff
+ self.Ax = self.A[:, :ndx]
+ self.Au = self.A[:, ndx:]
+ np.fill_diagonal(self.Au[2:, :], 1)
class ActuationModelFreeFloating:
'''
diff --git a/crocoddyl/flying.py b/crocoddyl/flying.py
new file mode 100644
index 0000000000000000000000000000000000000000..3a10c3e48fee2ec830d267c848e842695224ca22
--- /dev/null
+++ b/crocoddyl/flying.py
@@ -0,0 +1,79 @@
+import numpy as np
+import pinocchio
+
+from .differential_action import DifferentialActionDataAbstract, DifferentialActionModelAbstract
+from .state import StatePinocchio
+from .utils import a2m, m2a
+
+
+class DifferentialActionModelUAM(DifferentialActionModelAbstract):
+ def __init__(self, pinocchioModel, actuationModel, costModel):
+ DifferentialActionModelAbstract.__init__(self, pinocchioModel.nq, pinocchioModel.nv, actuationModel.nu)
+ self.DifferentialActionDataType = DifferentialActionDataUAM
+ self.pinocchio = pinocchioModel
+ self.State = StatePinocchio(self.pinocchio)
+ self.actuation = actuationModel
+ self.costs = costModel
+
+ @property
+ def ncost(self):
+ return self.costs.ncost
+
+ def calc(self, data, x, u=None):
+ if u is None:
+ u = self.unone
+ nq, nv = self.nq, self.nv
+ q = a2m(x[:nq])
+ v = a2m(x[-nv:])
+
+ data.tauq[:] = self.actuation.calc(data.actuation, x, u)
+
+ pinocchio.computeAllTerms(self.pinocchio, data.pinocchio, q, v)
+ data.M = data.pinocchio.M
+ data.Minv = np.linalg.inv(data.M)
+ # print data.Minv
+ # print data.tauq
+ # print data.pinocchio.nle
+ data.xout[:] = data.Minv * (a2m(data.tauq) - data.pinocchio.nle).flat
+
+ # --- Cost
+ pinocchio.forwardKinematics(self.pinocchio, data.pinocchio, q, v)
+ pinocchio.updateFramePlacements(self.pinocchio, data.pinocchio)
+ data.cost = self.costs.calc(data.costs, x, u)
+ return data.xout, data.cost
+
+ def calcDiff(self, data, x, u=None, recalc=True):
+ if u is None:
+ u = self.unone
+ if recalc:
+ xout, cost = self.calc(data, x, u)
+ nq, nv = self.nq, self.nv
+ q = a2m(x[:nq])
+ v = a2m(x[-nv:])
+ a = a2m(data.xout)
+
+ dtau_dx = data.actuation.Ax
+ dtau_du = data.actuation.Au
+
+ pinocchio.computeRNEADerivatives(self.pinocchio, data.pinocchio, q, v, a)
+ data.Fx[:, :nv] = -np.dot(data.Minv, data.pinocchio.dtau_dq)
+ data.Fx[:, nv:] = -np.dot(data.Minv, data.pinocchio.dtau_dv)
+ data.Fx += np.dot(data.Minv, dtau_dx)
+ data.Fu[:, :] = np.dot(data.Minv, dtau_du)
+ # Cost
+ pinocchio.computeJointJacobians(self.pinocchio, data.pinocchio, q)
+ pinocchio.updateFramePlacements(self.pinocchio, data.pinocchio)
+ self.costs.calcDiff(data.costs, x, u, recalc=False)
+ return data.xout, data.cost
+
+class DifferentialActionDataUAM(DifferentialActionDataAbstract):
+ def __init__(self, model):
+ self.pinocchio = model.pinocchio.createData()
+ self.actuation = model.actuation.createData(self.pinocchio)
+ costData = model.costs.createData(self.pinocchio)
+ # print self.costs.Lu
+ DifferentialActionDataAbstract.__init__(self, model, costData)
+
+ nv = model.nv
+ self.tauq = np.zeros(nv)
+ self.xout = np.zeros(nv)
diff --git a/crocoddyl/robots.py b/crocoddyl/robots.py
index 4b4c3774ac88074735f4d7d1283737f2721899d8..00d0e0c23aa07e04446fdb522b51d5be62881fea 100644
--- a/crocoddyl/robots.py
+++ b/crocoddyl/robots.py
@@ -125,3 +125,17 @@ def loadHyQ(modelPath='/opt/openrobots/share/example-robot-data'):
robot.q0[2] = 0.57750958
robot.model.referenceConfigurations["half_sitting"] = robot.q0
return robot
+
+def loadKinton(modelPath='/opt/openrobots/share/example-robot-data'):
+ URDF_FILENAME = "kinton_arm.urdf"
+ URDF_SUBPATH = "/kinton_description/urdf/" + URDF_FILENAME
+ robot = RobotWrapper.BuildFromURDF(modelPath + URDF_SUBPATH, [modelPath], pinocchio.JointModelFreeFlyer())
+ robot.q0.flat[7:] = [0, 0, 0, 0, 0, 0]
+ robot.model.referenceConfigurations["initial_pose"] = robot.q0
+ return robot
+
+def loadKintonArm(modelPath='/opt/openrobots/share/example-robot-data'):
+ URDF_FILENAME = "kinton_arm.urdf"
+ URDF_SUBPATH = "/kinton_description/urdf/" + URDF_FILENAME
+ robot = RobotWrapper.BuildFromURDF(modelPath + URDF_SUBPATH, [modelPath])
+ return robot
diff --git a/examples/kinton_flying.py b/examples/kinton_flying.py
new file mode 100644
index 0000000000000000000000000000000000000000..975a6fd8350235cd169093f0cf7bc43611af9e04
--- /dev/null
+++ b/examples/kinton_flying.py
@@ -0,0 +1,79 @@
+from crocoddyl import *
+import pinocchio as pin
+import numpy as np
+from crocoddyl.diagnostic import displayTrajectory
+
+# LOAD ROBOT
+robot = loadKinton()
+robot.initDisplay(loadModel=True)
+robot.display(robot.q0)
+
+robot.framesForwardKinematics(robot.q0)
+
+# DEFINE TARGET POSITION
+target_pos = np.array([0,0,3])
+target_quat = pin.Quaternion(0, 0, 0, 1)
+target_quat.normalize()
+
+# Plot goal frame
+robot.viewer.gui.addXYZaxis('world/framegoal', [1., 0., 0., 1.], .015, 4)
+robot.viewer.gui.applyConfiguration('world/framegoal', target_pos.tolist() + [target_quat[0], target_quat[1], target_quat[2], target_quat[3]])
+robot.viewer.gui.refresh()
+
+# ACTUATION MODEL
+actModel = ActuationModelUAM(robot.model)
+
+# COST MODEL
+# Create a cost model per the running and terminal action model.
+runningCostModel = CostModelSum(robot.model, actModel.nu)
+terminalCostModel = CostModelSum(robot.model, actModel.nu)
+
+frameName = 'base_link'
+state = StatePinocchio(robot.model)
+SE3ref = pin.SE3()
+SE3ref.translation = target_pos.reshape(3,1)
+SE3ref.rotation = target_quat.matrix()
+
+
+stateWeights = np.array([0] * 3 + [500.] * 3 + [0.01] * (robot.model.nv - 2) + [10] * robot.model.nv)
+goalTrackingCost = CostModelFramePlacement(robot.model,
+ frame=robot.model.getFrameId(frameName),
+ ref=SE3ref,
+ nu =actModel.nu)
+xRegCost = CostModelState(robot.model, state, ref=state.zero(), nu=actModel.nu)
+uRegCost = CostModelControl(robot.model, nu=robot.model.nv-2)
+
+# Then let's add the running and terminal cost functions
+runningCostModel.addCost(name="pos", weight=1e-3, cost=goalTrackingCost)
+runningCostModel.addCost(name="regx", weight=1e-7, cost=xRegCost)
+runningCostModel.addCost(name="regu", weight=1e-7, cost=uRegCost)
+terminalCostModel.addCost(name="pos", weight=50, cost=goalTrackingCost)
+
+# DIFFERENTIAL ACTION MODEL
+runningModel = IntegratedActionModelEuler(DifferentialActionModelUAM(robot.model, actModel, runningCostModel))
+terminalModel = IntegratedActionModelEuler(DifferentialActionModelUAM(robot.model, actModel, terminalCostModel))
+
+# DEFINING THE SHOOTING PROBLEM & SOLVING
+
+# Defining the time duration for running action models and the terminal one
+dt = 1e-3
+runningModel.timeStep = dt
+
+# For this optimal control problem, we define 250 knots (or running action
+# models) plus a terminal knot
+T = 250
+q0 = [0., 0., 0., 0., 0., 0.]
+q0 = robot.model.referenceConfigurations["initial_pose"]
+
+x0 = np.hstack([m2a(q0), np.zeros(robot.model.nv)])
+problem = ShootingProblem(x0, [runningModel] * T, terminalModel)
+
+# Creating the DDP solver for this OC problem, defining a logger
+ddp = SolverDDP(problem)
+ddp.callback = [CallbackDDPVerbose()]
+ddp.callback.append(CallbackDDPLogger())
+
+# Solving it with the DDP algorithm
+ddp.solve()
+
+displayTrajectory(robot, ddp.xs, runningModel.timeStep)
diff --git a/examples/notebooks/kinton_fixed_base.ipynb b/examples/notebooks/kinton_fixed_base.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..a5905634d6a2ef8914c18b5722e278e6ed8c1dd6
--- /dev/null
+++ b/examples/notebooks/kinton_fixed_base.ipynb
@@ -0,0 +1,1419 @@
+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "6\n"
+ ]
+ }
+ ],
+ "source": [
+ "from crocoddyl import *\n",
+ "import pinocchio as pin\n",
+ "import numpy as np\n",
+ "\n",
+ "robot = loadKintonArm()\n",
+ "robot.initDisplay(loadModel=True)\n",
+ "robot.display(robot.q0)\n",
+ "\n",
+ "robot.framesForwardKinematics(robot.q0)\n",
+ "\n",
+ "# Create a cost model per the running and terminal action model.\n",
+ "runningCostModel = CostModelSum(robot.model)\n",
+ "terminalCostModel = CostModelSum(robot.model)\n",
+ "\n",
+ "print robot.model.nv"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "target_pos = np.array([-0.1,-0.05,-0.15])\n",
+ "target_quat = pin.Quaternion(.4, .02, -.5, .7)\n",
+ "target_quat.normalize()\n",
+ "\n",
+ "# Plot goal frame\n",
+ "robot.viewer.gui.addXYZaxis('world/framegoal', [1., 0., 0., 1.], .015, 4)\n",
+ "robot.viewer.gui.applyConfiguration('world/framegoal', target_pos.tolist() + [target_quat[0], target_quat[1], target_quat[2], target_quat[3]])\n",
+ "robot.viewer.gui.refresh()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Note that we need to include a cost model (i.e. set of cost functions) in\n",
+ "# order to fully define the action model for our optimal control problem.\n",
+ "# For this particular example, we formulate three running-cost functions:\n",
+ "# goal-tracking cost, state and control regularization; and one terminal-cost:\n",
+ "# goal cost. First, let's create the common cost functions\n",
+ "frameName = 'link6'\n",
+ "state = StatePinocchio(robot.model)\n",
+ "SE3ref = pin.SE3()\n",
+ "SE3ref.translation = target_pos.reshape(3,1)\n",
+ "SE3ref.rotation = target_quat.matrix()\n",
+ "\n",
+ "goalTrackingCost = CostModelFramePlacement(robot.model, nu=robot.model.nv, frame=robot.model.getFrameId(frameName), ref=SE3ref)\n",
+ "xRegCost = CostModelState(robot.model, state, ref=state.zero(), nu=robot.model.nv)\n",
+ "uRegCost = CostModelControl(robot.model, nu=robot.model.nv)\n",
+ "\n",
+ "# Then let's add the running and terminal cost functions\n",
+ "runningCostModel.addCost(name=\"pos\", weight=1e-3, cost=goalTrackingCost)\n",
+ "runningCostModel.addCost(name=\"regx\", weight=1e-7, cost=xRegCost)\n",
+ "runningCostModel.addCost(name=\"regu\", weight=1e-7, cost=uRegCost)\n",
+ "terminalCostModel.addCost(name=\"pos\", weight=50, cost=goalTrackingCost)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "([array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.]),\n",
+ " array([-8.56312177e-04, 1.89532152e-03, 2.73421184e-03, 1.40535055e-04,\n",
+ " -6.80910601e-03, -1.47306350e-02, -8.56312177e-01, 1.89532152e+00,\n",
+ " 2.73421184e+00, 1.40535055e-01, -6.80910601e+00, -1.47306350e+01]),\n",
+ " array([-2.54926553e-03, 5.12874409e-03, 7.81062478e-03, -3.01186931e-05,\n",
+ " -1.90322663e-02, -4.14852078e-02, -1.69295335e+00, 3.23342257e+00,\n",
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+ " array([-1.78404033e-02, 2.30163561e-02, 4.41040294e-02, -8.14709277e-03,\n",
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+ " 1.07894509e+01, -3.35311661e+00, -2.38851526e+01, -5.43873060e+01]),\n",
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+ " array([-6.59052492e-02, 3.64822755e-02, 1.15410099e-01, -4.61780004e-02,\n",
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+ " 1.17993788e+01, -8.21606409e+00, -2.48046815e+01, -5.98022757e+01]),\n",
+ " array([-7.64316113e-02, 3.53798340e-02, 1.26909472e-01, -5.49984119e-02,\n",
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+ " 1.14993727e+01, -8.82041147e+00, -2.40053389e+01, -5.85164289e+01]),\n",
+ " array([-8.75234771e-02, 3.31730321e-02, 1.38043940e-01, -6.43585617e-02,\n",
+ " -3.00981066e-01, -7.00871445e-01, -1.10918658e+01, -2.20680192e+00,\n",
+ " 1.11344687e+01, -9.36014983e+00, -2.30618659e+01, -5.68792667e+01]),\n",
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+ " -3.22989193e-01, -7.55835190e-01, -1.15899167e+01, -3.32265166e+00,\n",
+ " 1.07236258e+01, -9.83713756e+00, -2.20081270e+01, -5.49637450e+01]),\n",
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+ " -3.43862344e-01, -8.08667742e-01, -1.20194151e+01, -4.43541893e+00,\n",
+ " 1.02827840e+01, -1.02539963e+01, -2.08731508e+01, -5.28325516e+01]),\n",
+ " array([-1.23513502e-01, 1.98823017e-02, 1.68875407e-01, -9.50635507e-02,\n",
+ " -3.63544010e-01, -8.59207080e-01, -1.23806934e+01, -5.53265979e+00,\n",
+ " 9.82505690e+00, -1.06138552e+01, -1.96816663e+01, -5.05393383e+01]),\n",
+ " array([-1.36188819e-01, 1.32784184e-02, 1.78236385e-01, -1.05983717e-01,\n",
+ " -3.81998590e-01, -9.07336913e-01, -1.26753164e+01, -6.60388340e+00,\n",
+ " 9.36097833e+00, -1.09201664e+01, -1.84545794e+01, -4.81298329e+01]),\n",
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+ " array([ 6.92989007, -2.26913873, 0.55662783, -1.23576312, -1.93313458,\n",
+ " -0.21961442]),\n",
+ " array([ 6.93668419, -2.00157024, 0.51924695, -1.25260927, -1.95976886,\n",
+ " -0.23393853]),\n",
+ " array([ 6.95343388, -1.73483135, 0.46752496, -1.27486597, -1.98815736,\n",
+ " -0.24279306]),\n",
+ " array([ 6.98065011, -1.46962003, 0.4013067 , -1.30324835, -2.01886515,\n",
+ " -0.24587415]),\n",
+ " array([ 7.01816458, -1.2066619 , 0.32107317, -1.33806248, -2.05215693,\n",
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+ " array([ 7.06518905, -0.94670995, 0.22788113, -1.37922765, -2.08801567,\n",
+ " -0.23449049]),\n",
+ " array([ 7.12041449, -0.69053767, 0.12326497, -1.42632031, -2.12617722,\n",
+ " -0.22039253]),\n",
+ " array([ 7.1821374 , -0.43892647, 0.00911335, -1.47863432, -2.16617627,\n",
+ " -0.20122747]),\n",
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+ " -0.05346534]),\n",
+ " array([ 7.57779833, 0.93462871, -0.75823523, -1.84478211, -2.4065301 ,\n",
+ " -0.01924376]),\n",
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+ " 0.01459886]),\n",
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+ " 0.04732453]),\n",
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+ " array([ 7.77408785, 1.84651103, -1.32943503, -2.11313677, -2.53941586,\n",
+ " 0.13277187]),\n",
+ " array([ 7.79073818, 2.00102316, -1.4268386 , -2.15574481, -2.55450954,\n",
+ " 0.1555189 ]),\n",
+ " array([ 7.80014695, 2.1464105 , -1.5185303 , -2.19432179, -2.56570724,\n",
+ " 0.17492016]),\n",
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+ " 0.1908433 ]),\n",
+ " array([ 7.79910886, 2.41064538, -1.68580281, -2.25973165, -2.57695032,\n",
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+ " array([ 7.73423073, 2.84079302, -1.96594454, -2.34925967, -2.56075374,\n",
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+ " array([ 7.70859457, 2.92971056, -2.02687623, -2.36456763, -2.55009688,\n",
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+ " array([ 7.65014557, 3.08753046, -2.14023544, -2.38860775, -2.52261241,\n",
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+ " 0.19282203]),\n",
+ " array([ 7.585453 , 3.22079917, -2.24408111, -2.40550785, -2.48839933,\n",
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+ " array([ 7.55182771, 3.27898277, -2.29298504, -2.41180844, -2.46926384,\n",
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+ " 0.15541443]),\n",
+ " array([ 7.48370293, 3.37983037, -2.38554968, -2.42095179, -2.42774946,\n",
+ " 0.14114163]),\n",
+ " array([ 7.44973117, 3.42299822, -2.42943559, -2.42405688, -2.40563681,\n",
+ " 0.12638359]),\n",
+ " array([ 7.41609605, 3.46163697, -2.47182966, -2.42632033, -2.38276689,\n",
+ " 0.11135912]),\n",
+ " array([ 7.38295241, 3.49595737, -2.51278289, -2.42781766, -2.35922815,\n",
+ " 0.09626413]),\n",
+ " array([ 7.35041715, 3.52615445, -2.55232757, -2.42860559, -2.33509393,\n",
+ " 0.08127107]),\n",
+ " array([ 7.31857256, 3.55240836, -2.59048052, -2.42872426, -2.3104242 ,\n",
+ " 0.06652895]),\n",
+ " array([ 7.28746963, 3.57488518, -2.62724598, -2.42819957, -2.28526743,\n",
+ " 0.05216377]),\n",
+ " array([ 7.25713136, 3.59373796, -2.66261848, -2.42704538, -2.25966237,\n",
+ " 0.03827945]),\n",
+ " array([ 7.22755588, 3.60910765, -2.69658527, -2.4252658 , -2.2336398 ,\n",
+ " 0.0249589 ]),\n",
+ " array([ 7.1987194 , 3.6211242 , -2.72912859, -2.42285721, -2.20722423,\n",
+ " 0.01226551]),\n",
+ " array([ 7.17057903e+00, 3.62990755e+00, -2.76022761e+00, -2.41981019e+00,\n",
+ " -2.18043539e+00, 2.44639452e-04]),\n",
+ " array([ 7.14307531, 3.63556857, -2.78986012, -2.41611129, -2.15328967,\n",
+ " -0.01107473]),\n",
+ " array([ 7.11613457, 3.63821002, -2.81800395, -2.41174457, -2.12580133,\n",
+ " -0.02167824]),\n",
+ " array([ 7.08967107, 3.63792742, -2.84463815, -2.40669295, -2.0979836 ,\n",
+ " -0.03156447]),\n",
+ " array([ 7.06358894, 3.6348098 , -2.86974393, -2.40093938, -2.06984958,\n",
+ " -0.0407432 ]),\n",
+ " array([ 7.03778386, 3.62894043, -2.8933054 , -2.39446783, -2.041413 ,\n",
+ " -0.04923377]),\n",
+ " array([ 7.01214456, 3.62039744, -2.91531013, -2.38726401, -2.01268885,\n",
+ " -0.05706344]),\n",
+ " array([ 6.98655419, 3.6092544 , -2.93574946, -2.37931603, -1.98369376,\n",
+ " -0.06426588]),\n",
+ " array([ 6.96089142, 3.59558075, -2.95461882, -2.37061484, -1.95444638,\n",
+ " -0.07087977]),\n",
+ " array([ 6.93503142, 3.57944223, -2.97191772, -2.36115447, -1.92496753,\n",
+ " -0.0769474 ]),\n",
+ " array([ 6.9088467 , 3.56090119, -2.98764979, -2.35093227, -1.89528025,\n",
+ " -0.08251349]),\n",
+ " array([ 6.88220779, 3.54001688, -3.00182265, -2.33994887, -1.86540979,\n",
+ " -0.08762401]),\n",
+ " array([ 6.85498381, 3.51684565, -3.01444772, -2.32820818, -1.83538349,\n",
+ " -0.09232516]),\n",
+ " array([ 6.82704292, 3.49144108, -3.02553994, -2.31571721, -1.80523053,\n",
+ " -0.0966624 ]),\n",
+ " array([ 6.79825267, 3.46385414, -3.0351175 , -2.30248584, -1.7749817 ,\n",
+ " -0.10067963]),\n",
+ " array([ 6.76848026, 3.43413325, -3.04320144, -2.28852653, -1.74466904,\n",
+ " -0.10441836]),\n",
+ " array([ 6.73759276, 3.40232429, -3.04981537, -2.27385399, -1.71432547,\n",
+ " -0.10791702]),\n",
+ " array([ 6.70545722, 3.36847069, -3.05498498, -2.25848479, -1.68398443,\n",
+ " -0.11121028]),\n",
+ " array([ 6.67194076, 3.33261338, -3.05873774, -2.24243694, -1.65367936,\n",
+ " -0.11432853]),\n",
+ " array([ 6.63691056, 3.29479078, -3.06110243, -2.22572949, -1.62344332,\n",
+ " -0.1172972 ]),\n",
+ " array([ 6.60023387, 3.25503878, -3.06210881, -2.20838207, -1.59330853,\n",
+ " -0.12013634]),\n",
+ " array([ 6.56177796, 3.21339071, -3.0617872 , -2.19041444, -1.56330586,\n",
+ " -0.12286004]),\n",
+ " array([ 6.52141004, 3.16987732, -3.06016812, -2.17184608, -1.53346441,\n",
+ " -0.12547599]),\n",
+ " array([ 6.47899715, 3.12452669, -3.05728193, -2.1526957 , -1.50381106,\n",
+ " -0.12798491]),\n",
+ " array([ 6.43440602, 3.07736426, -3.05315853, -2.13298089, -1.47437 ,\n",
+ " -0.13038011]),\n",
+ " array([ 6.38750293, 3.02841274, -3.04782698, -2.11271766, -1.44516235,\n",
+ " -0.13264689]),\n",
+ " array([ 6.33815354, 2.9776921 , -3.04131523, -2.09192006, -1.4162057 ,\n",
+ " -0.13476202]),\n",
+ " array([ 6.28622267, 2.92521956, -3.03364984, -2.07059983, -1.38751375,\n",
+ " -0.13669314]),\n",
+ " array([ 6.2315741 , 2.87100953, -3.02485566, -2.04876608, -1.35909593,\n",
+ " -0.13839805]),\n",
+ " array([ 6.1740703 , 2.81507362, -3.01495558, -2.02642492, -1.33095701,\n",
+ " -0.13982403]),\n",
+ " array([ 6.11357212, 2.7574206 , -3.00397025, -2.00357923, -1.3030968 ,\n",
+ " -0.14090704]),\n",
+ " array([ 6.0499385 , 2.69805643, -2.99191782, -1.98022842, -1.27550979,\n",
+ " -0.14157084]),\n",
+ " array([ 5.98302601, 2.6369842 , -2.9788136 , -1.95636817, -1.2481848 ,\n",
+ " -0.14172605]),\n",
+ " array([ 5.91268841, 2.57420417, -2.96466977, -1.93199036, -1.2211047 ,\n",
+ " -0.14126906]),\n",
+ " array([ 5.83877612, 2.50971375, -2.94949495, -1.9070829 , -1.194246 ,\n",
+ " -0.14008089]),\n",
+ " array([ 5.76113551, 2.44350746, -2.93329382, -1.88162977, -1.16757857,\n",
+ " -0.13802588]),\n",
+ " array([ 5.67960814, 2.375577 , -2.9160665 , -1.85561107, -1.14106512,\n",
+ " -0.13495032]),\n",
+ " array([ 5.59402975, 2.30591118, -2.89780788, -1.82900322, -1.11466083,\n",
+ " -0.13068087]),\n",
+ " array([ 5.50422913, 2.23449589, -2.87850674, -1.80177934, -1.08831271,\n",
+ " -0.12502289]),\n",
+ " array([ 5.41002664, 2.16131412, -2.85814457, -1.77390989, -1.06195892,\n",
+ " -0.11775858]),\n",
+ " array([ 5.31123248, 2.0863458 , -2.83669407, -1.7453635 , -1.03552781,\n",
+ " -0.10864502]),\n",
+ " array([ 5.20764456, 2.00956776, -2.81411711, -1.71610838, -1.00893678,\n",
+ " -0.09741197]),\n",
+ " array([ 5.0990458 , 1.9309534 , -2.79036211, -1.68611415, -0.98209066,\n",
+ " -0.08375949]),\n",
+ " array([ 4.98520101, 1.85047235, -2.76536047, -1.65535454, -0.95487959,\n",
+ " -0.06735541]),\n",
+ " array([ 4.86585291, 1.76808977, -2.73902177, -1.62381106, -0.92717618,\n",
+ " -0.04783259]),\n",
+ " array([ 4.74071739, 1.68376521, -2.71122737, -1.59147818, -0.89883163,\n",
+ " -0.02478587]),\n",
+ " array([ 4.60947781e+00, 1.59745075e+00, -2.68182175e+00, -1.55837035e+00,\n",
+ " -8.69670515e-01, 2.23110239e-03]),\n",
+ " array([ 4.47177814, 1.50908799, -2.65060074, -1.52453167, -0.83948367,\n",
+ " 0.0337094 ]),\n",
+ " array([ 4.32721496, 1.41860308, -2.61729554, -1.49004898, -0.80801854,\n",
+ " 0.07018862]),\n",
+ " array([ 4.1753282 , 1.325899 , -2.58155094, -1.45506975, -0.7749661 ,\n",
+ " 0.11226097]),\n",
+ " array([ 4.01559077, 1.2308432 , -2.5428955 , -1.41982605, -0.73994301,\n",
+ " 0.16057584]),\n",
+ " array([ 3.8473974 , 1.13324856, -2.50070069, -1.38466702, -0.70246751,\n",
+ " 0.2158447 ]),\n",
+ " array([ 3.67005332, 1.03284432, -2.45412485, -1.35010217, -0.66192663,\n",
+ " 0.27884666]),\n",
+ " array([ 3.48276368, 0.92923281, -2.40203646, -1.3168596 , -0.61753192,\n",
+ " 0.35043464]),\n",
+ " array([ 3.28462528, 0.82182713, -2.34291 , -1.28596362, -0.56825955,\n",
+ " 0.43154267]),\n",
+ " array([ 3.07462255, 0.70976532, -2.27468664, -1.25883928, -0.51276978,\n",
+ " 0.52319461]),\n",
+ " array([ 2.85163103, 0.59180017, -2.19459421, -1.237454 , -0.44929893,\n",
+ " 0.62651555]),\n",
+ " array([ 2.61443471, 0.46617382, -2.09892828, -1.22451277, -0.37551555,\n",
+ " 0.74274775]),\n",
+ " array([ 2.36177336, 0.33050882, -1.98281654, -1.22373052, -0.2883301 ,\n",
+ " 0.87327516]),\n",
+ " array([ 2.0924633 , 0.18179214, -1.84002523, -1.24020707, -0.18364463,\n",
+ " 1.01966444]),\n",
+ " array([ 1.80570754, 0.01661621, -1.66290677, -1.28089393, -0.05602076,\n",
+ " 1.18373752]),\n",
+ " array([ 1.50191892, -0.16796766, -1.44255224, -1.35498021, 0.10178605,\n",
+ " 1.36770709]),\n",
+ " array([ 1.18503942, -0.37207516, -1.16881095, -1.4736232 , 0.29960962,\n",
+ " 1.57444144]),\n",
+ " array([ 0.86953931, -0.58633231, -0.82798614, -1.64831962, 0.55190896,\n",
+ " 1.80801137]),\n",
+ " array([ 0.60030872, -0.77390263, -0.38975414, -1.89159215, 0.88828811,\n",
+ " 2.07472138]),\n",
+ " array([ 0.4764655 , -0.84946206, 0.22134384, -2.23923687, 1.44338066,\n",
+ " 2.37952862])],\n",
+ " True)"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "# Next, we need to create an action model for running and terminal knots. The\n",
+ "# forward dynamics (computed using ABA) are implemented\n",
+ "# inside DifferentialActionModelFullyActuated.\n",
+ "runningModel = IntegratedActionModelEuler(DifferentialActionModelFullyActuated(robot.model, runningCostModel))\n",
+ "terminalModel = IntegratedActionModelEuler(DifferentialActionModelFullyActuated(robot.model, terminalCostModel))\n",
+ "\n",
+ "# Defining the time duration for running action models and the terminal one\n",
+ "dt = 1e-3\n",
+ "runningModel.timeStep = dt\n",
+ "\n",
+ "# For this optimal control problem, we define 250 knots (or running action\n",
+ "# models) plus a terminal knot\n",
+ "T = 250\n",
+ "q0 = [0., 0., 0., 0., 0., 0.]\n",
+ "x0 = np.hstack([q0, np.zeros(robot.model.nv)])\n",
+ "problem = ShootingProblem(x0, [runningModel] * T, terminalModel)\n",
+ "\n",
+ "# Creating the DDP solver for this OC problem, defining a logger\n",
+ "ddp = SolverDDP(problem)\n",
+ "# ddp.callback = [CallbackDDPVerbose()]\n",
+ "# ddp.callback.append(CallbackDDPLogger())\n",
+ "\n",
+ "# Solving it with the DDP algorithm\n",
+ "ddp.solve()\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "from crocoddyl.diagnostic import displayTrajectory\n",
+ "\n",
+ "displayTrajectory(robot, ddp.xs, runningModel.timeStep)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 2",
+ "language": "python",
+ "name": "python2"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 2
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython2",
+ "version": "2.7.12"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/examples/notebooks/kinton_flying_base.ipynb b/examples/notebooks/kinton_flying_base.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..5a43b10c43988236c8380d18f8ee64b0ba27a80a
--- /dev/null
+++ b/examples/notebooks/kinton_flying_base.ipynb
@@ -0,0 +1,5251 @@
+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": 111,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "from crocoddyl import *\n",
+ "import pinocchio as pin\n",
+ "import numpy as np\n",
+ "from crocoddyl.diagnostic import displayTrajectory"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 112,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# LOAD ROBOT\n",
+ "robot = loadKinton()\n",
+ "robot.initDisplay(loadModel=True)\n",
+ "robot.display(robot.q0)\n",
+ "\n",
+ "robot.framesForwardKinematics(robot.q0)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 114,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# DEFINE TARGET POSITION\n",
+ "target_pos = np.array([0,0,1])\n",
+ "target_quat = pin.Quaternion(1, 0, 0, 0)\n",
+ "target_quat.normalize()\n",
+ "\n",
+ "# Plot goal frame\n",
+ "robot.viewer.gui.addXYZaxis('world/framegoal', [1., 0., 0., 1.], .015, 4)\n",
+ "robot.viewer.gui.applyConfiguration('world/framegoal', target_pos.tolist() + [target_quat[0], target_quat[1], target_quat[2], target_quat[3]])\n",
+ "robot.viewer.gui.refresh()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 115,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# ACTUATION MODEL\n",
+ "actModel = ActuationModelUAM(robot.model)\n",
+ "\n",
+ "# COST MODEL\n",
+ "# Create a cost model per the running and terminal action model.\n",
+ "runningCostModel = CostModelSum(robot.model, actModel.nu)\n",
+ "terminalCostModel = CostModelSum(robot.model, actModel.nu)\n",
+ "\n",
+ "frameName = 'base_link'\n",
+ "state = StatePinocchio(robot.model)\n",
+ "SE3ref = pin.SE3()\n",
+ "SE3ref.translation = target_pos.reshape(3,1)\n",
+ "SE3ref.rotation = target_quat.matrix()\n",
+ "\n",
+ "\n",
+ "stateWeights = np.array([0] * 3 + [500.] * 3 + [0.01] * (robot.model.nv - 2) + [10] * robot.model.nv)\n",
+ "goalTrackingCost = CostModelFramePlacement(robot.model,\n",
+ " frame=robot.model.getFrameId(frameName),\n",
+ " ref=SE3ref,\n",
+ " nu =actModel.nu)\n",
+ "xRegCost = CostModelState(robot.model, state, ref=state.zero(), nu=actModel.nu)\n",
+ "uRegCost = CostModelControl(robot.model, nu=robot.model.nv-2)\n",
+ "\n",
+ "# Then let's add the running and terminal cost functions\n",
+ "runningCostModel.addCost(name=\"pos\", weight=0.1, cost=goalTrackingCost)\n",
+ "runningCostModel.addCost(name=\"regx\", weight=1e-5, cost=xRegCost)\n",
+ "runningCostModel.addCost(name=\"regu\", weight=1e-7, cost=uRegCost)\n",
+ "terminalCostModel.addCost(name=\"pos\", weight=50, cost=goalTrackingCost)\n",
+ "\n",
+ "# DIFFERENTIAL ACTION MODEL\n",
+ "runningModel = IntegratedActionModelEuler(DifferentialActionModelUAM(robot.model, actModel, runningCostModel))\n",
+ "terminalModel = IntegratedActionModelEuler(DifferentialActionModelUAM(robot.model, actModel, terminalCostModel))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 116,
+ "metadata": {
+ "scrolled": true
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "iter \t cost \t stop \t grad \t xreg \t ureg \t step \t feas\n",
+ " 0 7.26954e+00 1.01735e-05 9.62434e+01 1.00000e-09 1.00000e-09 1.0000 1\n",
+ " 1 2.81986e+00 1.25848e-06 8.92713e+00 1.00000e-09 1.00000e-09 1.0000 1\n",
+ " 2 2.79917e+00 5.40348e-09 4.13316e-02 1.00000e-09 1.00000e-09 1.0000 1\n",
+ " 3 2.79916e+00 4.83151e-12 2.97873e-05 1.00000e-09 1.00000e-09 1.0000 1\n"
+ ]
+ },
+ {
+ "data": {
+ "text/plain": [
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+ " 6.09953860e-05, 1.02504859e-15]),\n",
+ " array([ 6.41110236e+00, -3.49979980e-15, -1.03159293e-04, 8.97824497e-16,\n",
+ " -9.41462565e-16, 2.00534662e-15, -1.85475726e-04, -2.51256580e-05,\n",
+ " 6.26362503e-05, 1.01416894e-15]),\n",
+ " array([ 6.19111875e+00, -3.62039128e-15, -1.10404991e-04, 8.78186839e-16,\n",
+ " -9.18508233e-16, 2.04422307e-15, -1.72251742e-04, -1.88842606e-05,\n",
+ " 6.41423531e-05, 1.00192650e-15]),\n",
+ " array([ 5.96838735e+00, -3.73209524e-15, -1.17393710e-04, 8.58314859e-16,\n",
+ " -8.95141492e-16, 2.07805075e-15, -1.59140217e-04, -1.27920808e-05,\n",
+ " 6.55088173e-05, 9.88281645e-16]),\n",
+ " array([ 5.74295448e+00, -3.83424655e-15, -1.24102302e-04, 8.38251890e-16,\n",
+ " -8.71412768e-16, 2.10657958e-15, -1.46153501e-04, -6.85699587e-06,\n",
+ " 6.67313389e-05, 9.73197339e-16]),\n",
+ " array([ 5.51487026e+00, -3.92612932e-15, -1.30504482e-04, 8.18020796e-16,\n",
+ " -8.47367354e-16, 2.12955450e-15, -1.33303605e-04, -1.08598459e-06,\n",
+ " 6.78065566e-05, 9.56640356e-16]),\n",
+ " array([ 5.28418868e+00, -4.00696520e-15, -1.36569848e-04, 7.97614645e-16,\n",
+ " -8.23042906e-16, 2.14671523e-15, -1.20601755e-04, 4.51546984e-06,\n",
+ " 6.87325867e-05, 9.38582955e-16]),\n",
+ " array([ 5.05096770e+00, -4.07589762e-15, -1.42262527e-04, 7.76984359e-16,\n",
+ " -7.98466466e-16, 2.15779563e-15, -1.08057744e-04, 9.94424094e-06,\n",
+ " 6.95097900e-05, 9.19005032e-16]),\n",
+ " array([ 4.81526938e+00, -4.13197060e-15, -1.47539286e-04, 7.56022424e-16,\n",
+ " -7.73650837e-16, 2.16252210e-15, -9.56789892e-05, 1.52007947e-05,\n",
+ " 7.01418692e-05, 8.97896822e-16]),\n",
+ " array([ 4.57715995e+00, -4.17410052e-15, -1.52346896e-04, 7.34541353e-16,\n",
+ " -7.48590145e-16, 2.16061036e-15, -8.34691825e-05, 2.02909572e-05,\n",
+ " 7.06374348e-05, 8.75262162e-16]),\n",
+ " array([ 4.33670994e+00, -4.20103868e-15, -1.56618434e-04, 7.12245265e-16,\n",
+ " -7.23254249e-16, 2.15175906e-15, -7.14263451e-05, 2.52284361e-05,\n",
+ " 7.10122357e-05, 8.51122125e-16]),\n",
+ " array([ 4.09399425e+00, -4.21132198e-15, -1.60268087e-04, 6.88692498e-16,\n",
+ " -6.97581515e-16, 2.13563773e-15, -5.95400512e-05, 3.00384141e-05,\n",
+ " 7.12923320e-05, 8.25518550e-16]),\n",
+ " array([ 3.84909224e+00, -4.20320850e-15, -1.63183830e-04, 6.63246701e-16,\n",
+ " -6.71469164e-16, 2.11186482e-15, -4.77874630e-05, 3.47626611e-05,\n",
+ " 7.15186048e-05, 7.98516356e-16]),\n",
+ " array([ 3.60208790e+00, -4.17459429e-15, -1.65217080e-04, 6.35013533e-16,\n",
+ " -6.44759944e-16, 2.07996913e-15, -3.61276760e-05, 3.94668040e-05,\n",
+ " 7.17531570e-05, 7.70202405e-16]),\n",
+ " array([ 3.35306987e+00, -4.12290705e-15, -1.66168066e-04, 6.02760076e-16,\n",
+ " -6.17222923e-16, 2.03932373e-15, -2.44936466e-05, 4.42506548e-05,\n",
+ " 7.20883804e-05, 7.40676502e-16]),\n",
+ " array([ 3.10213162e+00, -4.04497170e-15, -1.65765085e-04, 5.64815134e-16,\n",
+ " -5.88524279e-16, 1.98903716e-15, -1.27806266e-05, 4.92628697e-05,\n",
+ " 7.26597299e-05, 7.10025759e-16]),\n",
+ " array([ 2.84937149e+00, -3.93684043e-15, -1.63635057e-04, 5.18951907e-16,\n",
+ " -5.58179420e-16, 1.92778165e-15, -8.29397093e-07, 5.47217181e-05,\n",
+ " 7.36634863e-05, 6.78263878e-16]),\n",
+ " array([ 2.59489286e+00, -3.79357360e-15, -1.59261589e-04, 4.62263347e-16,\n",
+ " -5.25466477e-16, 1.85353938e-15, 1.15987696e-05, 6.09443994e-05,\n",
+ " 7.53806018e-05, 6.45193255e-16]),\n",
+ " array([ 2.33880424e+00, -3.60893628e-15, -1.51924727e-04, 3.91061030e-16,\n",
+ " -4.89252601e-16, 1.76326803e-15, 2.48621829e-05, 6.83880682e-05,\n",
+ " 7.82055520e-05, 6.10085765e-16]),\n",
+ " array([ 2.08121940e+00, -3.37491420e-15, -1.40612595e-04, 3.00878307e-16,\n",
+ " -4.47607214e-16, 1.65256381e-15, 3.95122943e-05, 7.77060182e-05,\n",
+ " 8.26690981e-05, 5.70911012e-16]),\n",
+ " array([ 1.82225748e+00, -3.08078125e-15, -1.23884834e-04, 1.86776559e-16,\n",
+ " -3.96865931e-16, 1.51561453e-15, 5.64594636e-05, 8.98201873e-05,\n",
+ " 8.94043159e-05, 5.22399943e-16]),\n",
+ " array([ 1.56204313e+00, -2.71098445e-15, -9.96366025e-05, 4.44576549e-17,\n",
+ " -3.29199412e-16, 1.34622204e-15, 7.74339700e-05, 1.05997569e-04,\n",
+ " 9.88569572e-05, 4.51152517e-16]),\n",
+ " array([ 1.30070665e+00, -2.23986866e-15, -6.46052190e-05, -1.26493664e-16,\n",
+ " -2.25906931e-16, 1.14145832e-15, 1.06473637e-04, 1.27853631e-04,\n",
+ " 1.09998533e-04, 3.23791818e-16]),\n",
+ " array([ 1.03838415e+00, -1.61795771e-15, -1.30755268e-05, -3.13540604e-16,\n",
+ " -3.78536677e-17, 9.09259840e-16, 1.55153805e-04, 1.56855941e-04,\n",
+ " 1.15373566e-04, 6.18545254e-17]),\n",
+ " array([ 7.75217670e-01, -7.35405379e-16, 6.72253889e-05, -4.65694376e-16,\n",
+ " 3.74459430e-16, 6.70714734e-16, 2.62210693e-04, 1.90324402e-04,\n",
+ " 8.35496516e-05, -4.92439235e-16]),\n",
+ " array([ 5.11355434e-01, 6.82544073e-16, 2.10097197e-04, -4.24062451e-16,\n",
+ " 1.43135020e-15, 3.80075570e-16, 5.75450582e-04, 1.85903667e-04,\n",
+ " -9.33411046e-05, -1.51716943e-15]),\n",
+ " array([ 2.46952099e-01, 3.46120206e-15, 5.34592006e-04, 2.17445316e-16,\n",
+ " 4.45149018e-15, -6.48985623e-16, 1.77986973e-03, -3.13205057e-04,\n",
+ " -6.45045565e-04, -2.45251105e-15])],\n",
+ " True)"
+ ]
+ },
+ "execution_count": 116,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "# DEFINING THE SHOOTING PROBLEM & SOLVING\n",
+ "\n",
+ "# Defining the time duration for running action models and the terminal one\n",
+ "dt = 1e-3\n",
+ "runningModel.timeStep = dt\n",
+ "\n",
+ "# For this optimal control problem, we define 250 knots (or running action\n",
+ "# models) plus a terminal knot\n",
+ "T = 500\n",
+ "q0 = robot.model.referenceConfigurations[\"initial_pose\"]\n",
+ "\n",
+ "x0 = np.hstack([m2a(q0), np.zeros(robot.model.nv)])\n",
+ "problem = ShootingProblem(x0, [runningModel] * T, terminalModel)\n",
+ "\n",
+ "# Creating the DDP solver for this OC problem, defining a logger\n",
+ "ddp = SolverDDP(problem)\n",
+ "ddp.callback = [CallbackDDPVerbose()]\n",
+ "ddp.callback.append(CallbackDDPLogger())\n",
+ "\n",
+ "# Solving it with the DDP algorithm\n",
+ "ddp.solve()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 118,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "displayTrajectory(robot, ddp.xs, runningModel.timeStep)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 119,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "thrust = []\n",
+ "Mr = [];\n",
+ "Mp = [];\n",
+ "My = [];\n",
+ "for u in ddp.us:\n",
+ " thrust.append(u[0])\n",
+ " Mr.append(u[1])\n",
+ " Mp.append(u[2])\n",
+ " My.append(u[3])"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 120,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "<matplotlib.legend.Legend at 0x7f2de8cf6050>"
+ ]
+ },
+ "execution_count": 120,
+ "metadata": {},
+ "output_type": "execute_result"
+ },
+ {
+ "data": {
+ "image/png": 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/Tt9NHEVEwpLsdSDvSnUh08Wp1cU89moTrUd6mVGUG3Y5IiIpk+xpvHnAh4H64Z9x97tO9JlstWR2KQCbmg9x6aLKkKsREUmdZC8k/AGJ03b7SYyBDL5khNNrEgGysflQyJWIiKRWsmMgde5+dUormSaqSvKoKsljU7POxBKR6S3ZI5D/MbOzUlrJNHJ6TamOQERk2hv1CMTM1pN4XnkM+ISZbSXxXBAjcVuss1NfYuY5vaaEX2zZT29/nNzYuG83JiKSEcbqwqrl6Cm8kqSlNaX0DThbWjqGxkRERKabsQLkLXffnpZKppGlNUfPxFKAiMh0NVaAzDKz20+00t3vOdG6bLagsojcWISNuw/xofPDrkZEJDXGCpAoUExizEOSFItGOK26hE17NJAuItPXWAHSrIsFJ2ZpTSlPb9qLu5N4iKOIyPQy1ilC+ptvgk6vKeHgkV72He4JuxQRkZQYK0CuSEsV09DQFem71Y0lItPTqAHi7gfTVch0c/oc3dJERKY3XeWWIqX5OcyvKGRdU1vYpYiIpERoAWJmc81slZltNLMNZvZnwfKZZva0mW0O3mcEy83Mvm5mjWa2zsym/Amy59SVs66pPewyRERSIswjkH7g0+6+FLgYuM3MlgJ3AM+6+2Lg2WAe4H3A4uC1Argv/SWPz9l1ZTS3d7PvcHfYpYiITLrQAsTdmwcfievuh4FNJG6dshx4KGj2EHB9ML0c+LYnvASUm1lNmssel3PmlgOwbqeOQkRk+pkSYyBmVg+cB7wMVLt7c7BqD1AdTNcCO4d9rClYNmWdMaeUiKFxEBGZlkIPEDMrBr4P/C93P+aUJXd3Es9eH8/2VpjZGjNb09IS7rPJC3NjnFpdwlqNg4jINBRqgJhZDonw+I67Px4s3jvYNRW87wuW7wLmDvt4XbDsGO5+v7s3uHtDVVVV6opP0tl1ZaxraiORhSIi00eYZ2EZ8CCwacRNGVcCtwTTt5B4nO7g8puDs7EuBtqHdXVNWWfXldPa2UdTa1fYpYiITKpkH2mbCpcBvwu8YWavB8v+D/Al4FEzuxXYDnw0WPcUcA3QCHQCn0hvuRNzTl1iIH1tUxtzZxaGXI2IyOQJLUDc/UVOfK+tt91CJRgPuS2lRaXAabNLyI1GWNfUzvvPnhN2OSIikyb0QfTpLjcW4fQ5pazdqTOxRGR6UYCkwbl1Zbyxq53+gXjYpYiITBoFSBpcUD+Tzt4BNjUfDrsUEZFJowBJgwvrZwCweptubiwi04cCJA1qygqoLS9gzXYFiIhMHwqQNLmwfgart7XqgkIRmTYUIGly4YKZtBzuYfuBzrBLERGZFAqQNLmwfiagcRARmT4UIGmyqKqYsoIc1mxrDbsUEZFJoQBJk0jEaJg/g9UaSBeRaUIBkkYN9TPZ2nKE/R09YZciInLSFCBptGxB4nqQNRoHEZFpQAGSRmfVllOQE+UXWw6EXYqIyElTgKRRbizCsgUzebFxf9iliIicNAVIml22qIItLUfY094ddikiIidFAZJmly2qBODnOgoRkQynAEmz02eXMrMol59vUYCISGZTgKRZJGJcckoFP2/cr/tiiUhGU4CE4B2LKtl7qIctLR1hlyIiMmEKkBBcfmoVAKvebAm5EhGRiVOAhKC2vIDTqkt47s19YZciIjJhCpCQvGvJLFZvO8ih7r6wSxERmRAFSEjevWQW/XHnxc06G0tEMlMs7AKy1fnzyikryOG5N/dxzVk1YZcjaTAQd7r6BujrjzPgzkD86Kt/2HTcnYgZEQML3hPzhlniTL7hy3KiRiwaISdq5EQiRCIW9q5KllCAhCQWjXD5qVU8/6t9xOOuP/QZ5EhPP83t3TS3d7G/o4fWI320dvYGrz5aj/RyuLufzt5+unoH6OwboLN3gN7+eFrqi0aMWMTIjUaIBeEyOJ0TjRCLJN5zjrMuJ2rEIsF8JEI0auREjGgksS4aSXwmFklMJ5YdXZcTiQRtEts52iax/cHaBrcRsUTbaMSImg19dmg6EiESYWhbsYjpz8oUknEBYmZXA18DosAD7v6lkEuasCtPn8UTa3fz2o5WGoInFkr43J29h3rY2tLBlv1H2NrSwfYDnexu62J3WxeHuvvf9hkzKC/IYUZhLjOKcqkozmVubgEFOTEKc6MU5kYpCN4H/xKPBH8hRiMRohGIRhLLDXAg7k7cE/XE3YnHE8vcj64biMfpG3D6g/e+gTj9wfvQ/Ih1vQNx+oet7+oboK97eJuj2xw8OuoP5hPv4V+/NBhgg6/YMdPHhs7wYIrY0bbHzid+g6HPDNtuZPj2zYhGg3kzcmMR8nOiQ6+CnCgFuRHyY1Hyc6PkxxK/e35OhIKgTV4sgtn0CMGMChAziwLfAN4DNAGrzWylu28Mt7KJefeSWeRGI/xo/R4FSEj6B+JsaTnC+l3trN/dzoZdh9jYfIiOnqMhUZATpb6yiLoZhVxYP5Oa8nzmlBVQU5ZPVUkeM4tyKc3PyZp/Gbsf2+02FC5DYRMfWtc3EA/eB9sOtoszEE8E4ECcobA6Xpfe4HzcE981EE90AfbHnYEBH+oO7I878eN+Nn7c7fb0xUdsd/i24sRH1NUff/v3TIQZFOfGKM6PUZx39L1kcD4vh+L8GCV5x7YpL8ihrCCH8sJcSvNjxKLhD2FnVIAAy4BGd98KYGbfA5YDGRkgJfk5XH5qJT96o5nPX3v6tPlXyVTW3TfA2p1tvPzWQV556yCvbm+lq28ASATF0jmlfOj8WhbNKmZhZTGnzCpidmm+fpthLOh2ikXDriRc7olg7O4foLt3gO6+xNFcd9/A0HviFR+2LE5Xbz8dPQN09PTR0dPP4e5+Onr62dPeTUdPPx3d/XT09jPWjSpK8mKUFQ6GSuK9rCB3aH7ezMKUj69mWoDUAjuHzTcBFw1vYGYrgBUA8+bNS19lE3T1mTU8s2kfa5vaOXduedjlTDvuzpaWDp57cx/PvbmP13a00dsfxwxOqy7how11nDdvBmfWlrKgspholhxFyMkzM3JjiW6s0vycSd12PO509g0kwqSnj/aufg519dHW1Ut7Zx9tXX20d/XR3pl4b+vqY++hDto6+2jv6qVvwDl/XrkCZLzc/X7gfoCGhobwO2vH8J7Tq4lFjB+tb1aATJJ43Hl1RytPrmvmuTf3seNgJwBLZpdw88XzuWhhBRfWz6C8MDfkSkWOLxKxoDsrBuSP67PuPnS0k2qZFiC7gLnD5uuCZRmrrDCHyxZV8uS6Zu64eom6SibI3Vm/6xBPrNvND9fuZnd7N3mxCJctquQPLl/Iu5fMora8IOwyRVLOzCjMjZGOfx9lWoCsBhab2QISwXEj8NvhlnTylp87h9sfXcvqba0sW6DB9PFo6+zl8dd28fArO9i8r4OcqHH54io+c/USrlxaHfwLTkRSIaP+dLl7v5l9CvgJidN4v+nuG0Iu66RddcZsCnPX8/hrTQqQJLg7r25v5f+9tJ2n1u+htz/OOXPL+dsPnsU1Z81W15RImmRUgAC4+1PAU2HXMZmK8mJcfeZsnlzXzBevO4P8nCw/veUE+gfi/HjDHv71Z2+xdmcbJXkxfqthLjcum8sZc8rCLk8k62RcgExXHz6/jsdf28XTG/fygXPmhF3OlNLdN8DDr+zgwRffoqm1i/kVhdy1/AxuuKCOwlz9LywSFv3pmyIuWVjBnLJ8/uPVJgVIoLtvgO++vIP7XthCy+EeGubP4PPXLuU9S6t1uq3IFKAAmSIiEeOjF87lq89sZtv+I9RXFoVdUmi6+wZ4ZPVOvrGqkX2He7h44Uz+6abzuHhhRdilicgw4V8LL0NuWjaPWMT4zsvbwy4lFPG48/hrTbz7H57nCys3UF9RxHf/4CK+t+IShYfIFKQjkCmkujSfq86YzaNrmrj9PadRkJs9g+kvbz3AXz+5iTd2tXNWbRl3f+QcLj2lQtfFiExhCpAp5ncvmc+TbzTzxNrdfPTCuWN/IMO9tf8If/fUJn66cS81Zfl85bfOYfk5tVlzY0KRTKYAmWIuWjCTJbNLuP9nW7nhgrpp+xdpb3+cf3lhC/euaiQnYvzFVadx6zsW6BRmkQyiMZApxsz443eeQuO+Dn66cW/Y5aTE6m0HuebrP+Oep3/Ne5dWs+rP38lt71qk8BDJMAqQKejas2qYX1HIfc834mPd0zmDtHf18bnH3+Aj//ILunoH+NYnLuTe3z6fWaXju1mciEwNCpApKBaN8IeXn8LapnZ+tnl/2OWcNHfnh+t2c+U9L/DI6h38wW8s4OnbL+ddp80KuzQROQkKkCnqwxfUUltewJd//CbxKfAI0Ylqau3k1ofW8Knv/pLZpfms/NQ7uPPapbqCXGQaUIBMUXmxKH9x1Wls2J24RXmm6R+I88DPtvKee/6bl7Ye4C/fv5T//OSlnFmre1aJTBf6Z+AUdt05c/jXn23l73/8K646Y3bGDDKv39XOHY+vY/2uQ1yxZBZ3XX+mnsUhMg3pCGQKi0SMO685nV1tXfzzqsawyxnTkZ5+/vqHG7nu3hfZe6iHf/7Y+TxwS4PCQ2Sa0hHIFHfpoko+eF4t972whfefM4dTq0vCLum4nntzL3/5XxvY1dbFxy6ax2euXkJZweQ+J1pEphYdgWSAz197OsV5Me74/jr6B1L/nOPx2He4m9u++xq/929rKMqL8v0/voS/+eBZCg+RLKAAyQAVxXl88bozeG1HG19/dnPY5QAwEHf+/aXtXPGPL/D0xr38+XtP5Yd/8htcMF9PVBTJFurCyhDLz63lxc37+adVjSxbUME7FleGVsv6Xe3c+V/rWbuzjUtPqeBvPngWC7L49vMi2UpHIBnkruVnsnhWMX/y8GtsaelI+/cf7u7jiys3cN29L7KrtZOv3Xgu3/n9ixQeIllKAZJBCnKj/OvNDUQjxs0PvsLeQ91p+d6BuPMfa3ZyxT++wEO/2MbHLprPs59+J8vPrdXt1kWymAIkw8yvKOJbH19Ga2cvN93/Ek2tnSn9vp837ucD//Qif/HYOmrK8vnPT17GX11/pgbJRUQBkonOqivjod9bxv6OHj70z//D+l3tk/4dr24/yM3ffIWPPfAy7V19fP2m8/jPT17GuXPLJ/27RCQz2XS62+tIDQ0NvmbNmrDLSJlf7TnMx7/1Cvs7evjMVUu49R0LTur5Ie7OzxsP8I1Vjfxi6wFmFuXyh5cv5JZL6zPmKngROXlm9qq7N4zZTgGS2VqP9PLZ76/jpxv3cnpNKZ+9+jR+89SqcY1N7Gnv5om1u3n4lR1s3X+EWSV5rLh8Ib990Tzd9FAkC03pADGzu4EPAL3AFuAT7t4WrPsccCswAPypu/8kWH418DUgCjzg7l8a63uyIUAgceTwxLpm7v7Jm+w82MXCyiI+eF4tly6q5MzaUvJi0WPatnX2sbapjdd3tvH8r1p4fWcbAA3zZ/Cxi+fxvjNrdMQhksWmeoC8F3jO3fvN7MsA7v5ZM1sKPAwsA+YAzwCnBh/7NfAeoAlYDdzk7htH+55sCZBBPf0DPLG2mUdX7+SVbQeHllcW51KUFyPuzsGOXo70DgBgBmfOKePqM2dz1RmzWTSrOKzSRWQKSTZAQumfcPefDpt9CbghmF4OfM/de4C3zKyRRJgANLr7VgAz+17QdtQAyTZ5sSg3XFDHDRfUsb+jh9VvHeTXeztobu+iuy8RGjOL8qgpy+eMOaWcVVdGSb7OphKRiZkKHdy/BzwSTNeSCJRBTcEygJ0jll+U+tIyV2VxHu87q4b3nRV2JSIyXaUsQMzsGWD2cVbd6e4/CNrcCfQD35nE710BrACYN2/eZG1WRERGSFmAuPuVo603s48D7weu8KMDMbuAucOa1QXLGGX5yO+9H7gfEmMg4y5cRESSEsqFhMEZVZ8BrnP34ZdSrwRuNLM8M1sALAZeITFovtjMFphZLnBj0FZEREIS1hjIvUAe8HRwvcJL7v5H7r7BzB4lMTjeD9zm7gMAZvYp4CckTuP9prtvCKepPu1XAAADeElEQVR0EREBXUgoIiIjJHsar+6FJSIiE6IAERGRCVGAiIjIhEzrMRAzawG2n8QmKoH9k1ROptA+Zwftc3aY6D7Pd/eqsRpN6wA5WWa2JpmBpOlE+5wdtM/ZIdX7rC4sERGZEAWIiIhMiAJkdPeHXUAItM/ZQfucHVK6zxoDERGRCdERiIiITEjWB4iZXW1mvzKzRjO74zjr88zskWD9y2ZWn/4qJ1cS+3y5mb1mZv1mdsPxtpFpktjn281so5mtM7NnzWx+GHVOpiT2+Y/M7A0ze93MXgyeCJrRxtrnYe0+bGZuZhl/VlYSv/PHzawl+J1fN7Pfn7Qvd/esfZG4MeMWYCGQC6wFlo5o80ngX4LpG4FHwq47DftcD5wNfBu4Ieya07TP7wIKg+k/zpLfuXTY9HXAj8OuO9X7HLQrAf6bxMPrGsKuOw2/88eBe1Px/dl+BLKM4FG57t4LDD4qd7jlwEPB9GPAFRbcQjhDjbnP7r7N3dcB8TAKTIFk9nmVH320wEsknjmTyZLZ50PDZouATB8QTebPM8BfAV8GutNZXIoku88pke0BUsvbH5Vbe6I27t4PtAMVaakuNZLZ5+lmvPt8K/CjlFaUeknts5ndZmZbgL8H/jRNtaXKmPtsZucDc939yXQWlkLJ/r/94aB79jEzm3uc9ROS7QEicgwz+x2gAbg77FrSwd2/4e6nAJ8FPh92PalkZhHgHuDTYdeSZk8A9e5+NvA0R3tUTlq2B8hoj9B9WxsziwFlwIG0VJcayezzdJPUPpvZlcCdJJ6U2ZOm2lJlvL/z94DrU1pR6o21zyXAmcDzZrYNuBhYmeED6WP+zu5+YNj/zw8AF0zWl2d7gCTzqNyVwC3B9A3Acx6MTGWobHw88Jj7bGbnAf+XRHjsC6HGyZbMPi8eNnstsDmN9aXCqPvs7u3uXunu9e5eT2Ks6zp3z+SnziXzO9cMm70O2DRp3x72WQRhv4BrgF+TOJPhzmDZXST+xwLIB/4DaCTxfPaFYdechn2+kERf6hESR1sbwq45Dfv8DLAXeD14rQy75jTs89eADcH+rgLOCLvmVO/ziLbPk+FnYSX5O/9d8DuvDX7nJZP13boSXUREJiTbu7BERGSCFCAiIjIhChAREZkQBYiIiEyIAkRERCZEASIiIhOiABERkQlRgIiIyIT8f42ZWAox6qx8AAAAAElFTkSuQmCC\n",
+ "text/plain": [
+ "<matplotlib.figure.Figure at 0x7f2deb729d90>"
+ ]
+ },
+ "metadata": {
+ "needs_background": "light"
+ },
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
+ "<matplotlib.figure.Figure at 0x7f2df03da850>"
+ ]
+ },
+ "metadata": {
+ "needs_background": "light"
+ },
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "import matplotlib.pyplot as plt\n",
+ "t = np.arange(0., T*dt, dt)\n",
+ "\n",
+ "plt.figure()\n",
+ "plt.plot(t, thrust)\n",
+ "plt.ylabel('Thrust, [N]')\n",
+ "\n",
+ "plt.figure()\n",
+ "plt.plot(t,Mr, t,Mp, t,My)\n",
+ "plt.legend(['Mr','Mp','My'])"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 92,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "robot.display(q0)"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 2",
+ "language": "python",
+ "name": "python2"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 2
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython2",
+ "version": "2.7.12"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}