This is an implementation of the Iterative Linear Quadratic Regulator (iLQR) for non-linear trajectory optimization based on Yuval Tassa's paper.
It is compatible with both Python 2 and 3 and has built-in support for auto-differentiating both the dynamics model and the cost function using Theano.
To install, clone and run:
python setup.py installYou may also install the dependencies with pipenv as follows:
pipenv installAfter installing, import as follows:
fromilqrimportiLQRYou can see the examples directory for Jupyter notebooks to see how common control problems can be solved through iLQR.
You can set up your own dynamics model by either extending the Dynamics
class and hard-coding it and its partial derivatives. Alternatively, you can
write it up as a Theano expression and use the AutoDiffDynamics class
for it to be auto-differentiated. Finally, if all you have is a function, you
can use the FiniteDiffDynamics class to approximate the derivatives
with finite difference approximation.
This section demonstrates how to implement the following dynamics model:
m \dot{v} = F - \alpha v
where m is the object's mass in kg, alpha is the friction coefficient, v is the object's velocity in m/s, \dot{v} is the object's acceleration in m/s^2, and F is the control (or force) you're applying to the object in N.
importtheano.tensorasTfromilqr.dynamicsimportAutoDiffDynamicsx=T.dscalar("x") # Position.x_dot=T.dscalar("x_dot") # Velocity.F=T.dscalar("F") # Force.dt=0.01# Discrete time-step in seconds.m=1.0# Mass in kg.alpha=0.1# Friction coefficient.# Acceleration.x_dot_dot=x_dot* (1-alpha*dt/m) +F*dt/m# Discrete dynamics model definition.f=T.stack([
x+x_dot*dt,
x_dot+x_dot_dot*dt,
])
x_inputs= [x, x_dot] # State vector.u_inputs= [F] # Control vector.# Compile the dynamics.# NOTE: This can be slow as it's computing and compiling the derivatives.# But that's okay since it's only a one-time cost on startup.dynamics=AutoDiffDynamics(f, x_inputs, u_inputs)Note: If you want to be able to use the Hessians (f_xx, f_ux,
and f_uu), you need to pass the hessians=True argument to the
constructor. This will increase compilation time. Note that iLQR does
not require second-order derivatives to function.
importtheano.tensorasTfromilqr.dynamicsimportBatchAutoDiffDynamicsstate_size=2# [position, velocity]action_size=1# [force]dt=0.01# Discrete time-step in seconds.m=1.0# Mass in kg.alpha=0.1# Friction coefficient.deff(x, u, i):
"""Batched implementation of the dynamics model. Args: x: State vector [*, state_size]. u: Control vector [*, action_size]. i: Current time step [*, 1]. Returns: Next state vector [*, state_size]. """x_=x[..., 0]
x_dot=x[..., 1]
F=u[..., 0]
# Acceleration.x_dot_dot=x_dot* (1-alpha*dt/m) +F*dt/m# Discrete dynamics model definition.returnT.stack([
x_+x_dot*dt,
x_dot+x_dot_dot*dt,
]).T# Compile the dynamics.# NOTE: This can be slow as it's computing and compiling the derivatives.# But that's okay since it's only a one-time cost on startup.dynamics=BatchAutoDiffDynamics(f, state_size, action_size)Note: This is a faster version of AutoDiffDynamics that doesn't
support Hessians.
fromilqr.dynamicsimportFiniteDiffDynamicsstate_size=2# [position, velocity]action_size=1# [force]dt=0.01# Discrete time-step in seconds.m=1.0# Mass in kg.alpha=0.1# Friction coefficient.deff(x, u, i):
"""Dynamics model function. Args: x: State vector [state_size]. u: Control vector [action_size]. i: Current time step. Returns: Next state vector [state_size]. """
[x, x_dot] =x
[F] =u# Acceleration.x_dot_dot=x_dot* (1-alpha*dt/m) +F*dt/mreturnnp.array([
x+x_dot*dt,
x_dot+x_dot_dot*dt,
])
# NOTE: Unlike with AutoDiffDynamics, this is instantaneous, but will not be# as accurate.dynamics=FiniteDiffDynamics(f, state_size, action_size)Note: It is possible you might need to play with the epsilon values
(x_eps and u_eps) used when computing the approximation if you
run into numerical instability issues.
Regardless of the method used for constructing your dynamics model, you can use them as follows:
curr_x=np.array([1.0, 2.0])
curr_u=np.array([0.0])
i=0# This dynamics model is not time-varying, so this doesn't matter.>>>dynamics.f(curr_x, curr_u, i)
... array([ 1.02 , 2.01998])
>>>dynamics.f_x(curr_x, curr_u, i)
... array([[ 1. , 0.01 ],
[ 0. , 1.00999]])
>>>dynamics.f_u(curr_x, curr_u, i)
... array([[ 0. ],
[ 0.0001]])Comparing the output of the AutoDiffDynamics and the
FiniteDiffDynamics models should generally yield consistent results,
but the auto-differentiated method will always be more accurate. Generally, the
finite difference approximation will be faster unless you're also computing the
Hessians: in which case, Theano's compiled derivatives are more optimized.
Similarly, you can set up your own cost function by either extending the
Cost class and hard-coding it and its partial derivatives.
Alternatively, you can write it up as a Theano expression and use the
AutoDiffCost class for it to be auto-differentiated. Finally, if all
you have are a loss functions, you can use the FiniteDiffCost class to
approximate the derivatives with finite difference approximation.
The most common cost function is the quadratic format used by Linear Quadratic Regulators:
(x - x_{goal})^T Q (x - x_{goal}) + (u - u_{goal})^T R (u - u_{goal})
where Q and R are matrices defining your quadratic state error
and quadratic control errors and x_{goal} is your target state. For
convenience, an implementation of this cost function is made available as the
QRCost class.
importnumpyasnpfromilqr.costimportQRCoststate_size=2# [position, velocity]action_size=1# [force]# The coefficients weigh how much your state error is worth to you vs# the size of your controls. You can favor a solution that uses smaller# controls by increasing R's coefficient.Q=100*np.eye(state_size)
R=0.01*np.eye(action_size)
# This is optional if you want your cost to be computed differently at a# terminal state.Q_terminal=np.array([[100.0, 0.0], [0.0, 0.1]])
# State goal is set to a position of 1 m with no velocity.x_goal=np.array([1.0, 0.0])
# NOTE: This is instantaneous and completely accurate.cost=QRCost(Q, R, Q_terminal=Q_terminal, x_goal=x_goal)importtheano.tensorasTfromilqr.costimportAutoDiffCostx_inputs= [T.dscalar("x"), T.dscalar("x_dot")]
u_inputs= [T.dscalar("F")]
x=T.stack(x_inputs)
u=T.stack(u_inputs)
x_diff=x-x_goall=x_diff.T.dot(Q).dot(x_diff) +u.T.dot(R).dot(u)
l_terminal=x_diff.T.dot(Q_terminal).dot(x_diff)
# Compile the cost.# NOTE: This can be slow as it's computing and compiling the derivatives.# But that's okay since it's only a one-time cost on startup.cost=AutoDiffCost(l, l_terminal, x_inputs, u_inputs)importtheano.tensorasTfromilqr.costimportBatchAutoDiffCostdefcost_function(x, u, i, terminal):
"""Batched implementation of the quadratic cost function. Args: x: State vector [*, state_size]. u: Control vector [*, action_size]. i: Current time step [*, 1]. terminal: Whether to compute the terminal cost. Returns: Instantaneous cost [*]. """Q_=Q_terminalifterminalelseQl=x.dot(Q_).dot(x.T)
ifl.ndim==2:
l=T.diag(l)
ifnotterminal:
l_u=u.dot(R).dot(u.T)
ifl_u.ndim==2:
l_u=T.diag(l_u)
l+=l_ureturnl# Compile the cost.# NOTE: This can be slow as it's computing and compiling the derivatives.# But that's okay since it's only a one-time cost on startup.cost=BatchAutoDiffCost(cost_function, state_size, action_size)fromilqr.costimportFiniteDiffCostdefl(x, u, i):
"""Instantaneous cost function. Args: x: State vector [state_size]. u: Control vector [action_size]. i: Current time step. Returns: Instantaneous cost [scalar]. """x_diff=x-x_goalreturnx_diff.T.dot(Q).dot(x_diff) +u.T.dot(R).dot(u)
defl_terminal(x, i):
"""Terminal cost function. Args: x: State vector [state_size]. i: Current time step. Returns: Terminal cost [scalar]. """x_diff=x-x_goalreturnx_diff.T.dot(Q_terminal).dot(x_diff)
# NOTE: Unlike with AutoDiffCost, this is instantaneous, but will not be as# accurate.cost=FiniteDiffCost(l, l_terminal, state_size, action_size)Note: It is possible you might need to play with the epsilon values
(x_eps and u_eps) used when computing the approximation if you
run into numerical instability issues.
Regardless of the method used for constructing your cost function, you can use them as follows:
>>>cost.l(curr_x, curr_u, i)
... 400.0>>>cost.l_x(curr_x, curr_u, i)
... array([ 0., 400.])
>>>cost.l_u(curr_x, curr_u, i)
... array([ 0.])
>>>cost.l_xx(curr_x, curr_u, i)
... array([[ 200., 0.],
[ 0., 200.]])
>>>cost.l_ux(curr_x, curr_u, i)
... array([[ 0., 0.]])
>>>cost.l_uu(curr_x, curr_u, i)
... array([[ 0.02]])N=1000# Number of time-steps in trajectory.x0=np.array([0.0, -0.1]) # Initial state.us_init=np.random.uniform(-1, 1, (N, 1)) # Random initial action path.ilqr=iLQR(dynamics, cost, N)
xs, us=ilqr.fit(x0, us_init)xs and us now hold the optimal state and control trajectory
that reaches the desired goal state with minimum cost.
Finally, a RecedingHorizonController is also bundled with this package
to use the iLQR controller in Model Predictive Control.
To quote from Tassa's paper: "Two important parameters which have a direct
impact on performance are the simulation time-step dt and the horizon
length N. Since speed is of the essence, the goal is to choose those
values which minimize the number of steps in the trajectory, i.e. the largest
possible time-step and the shortest possible horizon. The size of dt
is limited by our use of Euler integration; beyond some value the simulation
becomes unstable. The minimum length of the horizon N is a
problem-dependent quantity which must be found by trial-and-error."
Contributions are welcome. Simply open an issue or pull request on the matter.
We use YAPF for all Python formatting needs. You can auto-format your changes with the following command:
yapf --recursive --in-place --parallel .You may install the linter as follows:
pipenv install --devSee LICENSE.
This implementation was partially based on Yuval Tassa's MATLABimplementation,
and navigator8972's
implementation.