simple_pendulum.controllers.lqr.roa

Submodules

simple_pendulum.controllers.lqr.roa.plot

simple_pendulum.controllers.lqr.roa.plot.get_ellipse_params(rho, M): Returns ellipse params (excl center point)

simple_pendulum.controllers.lqr.roa.plot.get_ellipse_patch(px, py, rho, M, alpha_val=1, linec='red', facec='none', linest='solid'): return an ellipse patch

simple_pendulum.controllers.lqr.roa.plot.plot_ellipse(px, py, rho, M, save_to=None, show=True)

simple_pendulum.controllers.lqr.roa.sampling

simple_pendulum.controllers.lqr.roa.sampling.najafi_based_sampling(plant, controller, n=10000, rho0=100, M=None, x_star=array([3.14159265, 0.0]))

Estimate the RoA for the closed loop dynamics using the method introduced in Najafi, E., Babuška, R. & Lopes, G.A.D. A fast sampling method for estimating the domain of attraction. Nonlinear Dyn 86, 823–834 (2016). https://doi.org/10.1007/s11071-016-2926-7

Parameters:

plantsimple_pendulum.model.pendulum_plant: configured pendulum plant object
controllersimple_pendulum.controllers.lqr.lqr_controller: configured lqr controller object
nint, optional: number of samples, by default 100000
rho0int, optional: initial estimate of rho, by default 10
Mnp.array, optional: M, such that x_barT M x_bar is the Lyapunov fct. by default None, and controller.S is used
x_starnp.array, optional: nominal position (fixed point of the nonlinear dynamics)

Returns:

rhofloat: estimated value of rho
Mnp.array: M

simple_pendulum.controllers.lqr.roa.sos

simple_pendulum.controllers.lqr.roa.sos.SOSequalityConstrained(pendulum, controller)

Estimate the RoA for the closed loop dynamics using the method described by Russ Tedrake in “Underactuated Robotics: Algorithms for Walking, Running, Swimming, Flying, and Manipulation”,: Course Notes for MIT 6.832, 2022, “http://underactuated.mit.edu”, sec. 9.3.2: “The equality-constrained formulation”. This is discussed a bit more by Shen Shen and Russ Tedrake in “Sampling Quotient-Ring Sum-of-Squares Programs for Scalable Verification of Nonlinear Systems”, Proceedings of the 2020 59th IEEE Conference on Decision and Control (CDC) , 2020., http://groups.csail.mit.edu/robotics-center/public_papers/Shen20.pdf, pg. 2-3.

Parameters:

pendulumsimple_pendulum.model.pendulum_plant: configured pendulum plant object
controllersimple_pendulum.controllers.lqr.lqr_controller: configured lqr controller object

Returns:

rhofloat: estimated value of rho
Snp.array: S matrix from the lqr controller

simple_pendulum.controllers.lqr.roa.sos.SOSlineSearch(pendulum, controller)

Simple line search(bisection method) on rho that search for the maximum rho which satisfies the Lyapunov conditions in order to obtain an estimate of the RoA for the closed loop dynamics.

Parameters:

pendulumsimple_pendulum.model.pendulum_plant: configured pendulum plant object
controllersimple_pendulum.controllers.lqr.lqr_controller: configured lqr controller object

Returns:

rhofloat: estimated value of rho
Snp.array: S matrix from the lqr controller

simple_pendulum.controllers.lqr.roa.utils

class simple_pendulum.controllers.lqr.roa.utils.PendulumPlantApprox(mass=1.0, length=0.5, damping=0.1, gravity=9.81, coulomb_fric=0.0, inertia=None, torque_limit=inf, taylorApprox_order=1)

Bases: object

forward_dynamics(state, tau)

Computes forward dynamics

Parameters:

statearray like: len(state)=2 The state of the pendulum [angle, angular velocity] floats, units: rad, rad/s
taufloat: motor torque, unit: Nm

Returns:

float, angular acceleration, unit: rad/s^2

rhs(t, state, tau)

Computes the integrand of the equations of motion.

Parameters:

tfloat: time, not used (the dynamics of the pendulum are time independent)
statearray like: len(state)=2 The state of the pendulum [angle, angular velocity] floats, units: rad, rad/s
taufloat or array like: motor torque, unit: Nm

Returns:

resarray like: the integrand, contains [angular velocity, angular acceleration]

simple_pendulum.controllers.lqr.roa.utils.direct_sphere(d, r_i=0, r_o=1)

Direct Sampling from the d Ball based on Krauth, Werner. Statistical Mechanics: Algorithms and Computations. Oxford Master Series in Physics 13. Oxford: Oxford University Press, 2006. page 42

Parameters:

dint: dimension of the ball
r_iint, optional: inner radius, by default 0
r_oint, optional: outer radius, by default 1

Returns:

np.array: random vector directly sampled from the solid d Ball

simple_pendulum.controllers.lqr.roa.utils.quad_form(M, x): Helper function to compute quadratic forms such as x^TMx

simple_pendulum.controllers.lqr.roa.utils.rhoVerification(rho, pendulum, controller)

SOS Verification of the Lyapunov conditions for a given rho in order to obtain an estimate of the RoA for the closed loop dynamics.: This method is described by Russ Tedrake in “Underactuated Robotics: Algorithms for Walking, Running, Swimming, Flying, and Manipulation”, Course Notes for MIT 6.832, 2022, “http://underactuated.mit.edu”, sec. 9.3.2: “Basic region of attraction formulation”.

Parameters:

rho: float: value of rho to be verified
pendulumsimple_pendulum.model.pendulum_plant: configured pendulum plant object
controllersimple_pendulum.controllers.lqr.lqr_controller: configured lqr controller object

Returns:

resultboolean: result of the verification

simple_pendulum.controllers.lqr.roa.utils.sample_from_ellipsoid(M, rho, r_i=0, r_o=1)

sample directly from the ellipsoid defined by xT M x.

Parameters:

Mnp.array: Matrix M such that xT M x leq rho defines the hyperellipsoid to sample from
rhofloat: rho such that xT M x leq rho defines the hyperellipsoid to sample from
r_iint, optional: inner radius, by default 0
r_oint, optional: outer radius, by default 1

Returns:

np.array: random vector from within the hyperellipsoid

simple_pendulum.controllers.lqr.roa.utils.vol_ellipsoid(rho, M): Calculate the Volume of a Hyperellipsoid Volume of the Hyperllipsoid according to https://math.stackexchange.com/questions/332391/volume-of-hyperellipsoid/332434 Intuition: https://textbooks.math.gatech.edu/ila/determinants-volumes.html Volume of n-Ball https://en.wikipedia.org/wiki/Volume_of_an_n-ball