28 research outputs found
Reconstruction of Support of a Measure From Its Moments
In this paper, we address the problem of reconstruction of support of a
measure from its moments. More precisely, given a finite subset of the moments
of a measure, we develop a semidefinite program for approximating the support
of measure using level sets of polynomials. To solve this problem, a sequence
of convex relaxations is provided, whose optimal solution is shown to converge
to the support of measure of interest. Moreover, the provided approach is
modified to improve the results for uniform measures. Numerical examples are
presented to illustrate the performance of the proposed approach.Comment: This has been submitted to the 53rd IEEE Conference on Decision and
Contro
Convex Chance Constrained Model Predictive Control
We consider the Chance Constrained Model Predictive Control problem for
polynomial systems subject to disturbances. In this problem, we aim at finding
optimal control input for given disturbed dynamical system to minimize a given
cost function subject to probabilistic constraints, over a finite horizon. The
control laws provided have a predefined (low) risk of not reaching the desired
target set. Building on the theory of measures and moments, a sequence of
finite semidefinite programmings are provided, whose solution is shown to
converge to the optimal solution of the original problem. Numerical examples
are presented to illustrate the computational performance of the proposed
approach.Comment: This work has been submitted to the 55th IEEE Conference on Decision
and Contro
Moment-Sum-Of-Squares Approach For Fast Risk Estimation In Uncertain Environments
In this paper, we address the risk estimation problem where one aims at
estimating the probability of violation of safety constraints for a robot in
the presence of bounded uncertainties with arbitrary probability distributions.
In this problem, an unsafe set is described by level sets of polynomials that
is, in general, a non-convex set. Uncertainty arises due to the probabilistic
parameters of the unsafe set and probabilistic states of the robot. To solve
this problem, we use a moment-based representation of probability
distributions. We describe upper and lower bounds of the risk in terms of a
linear weighted sum of the moments. Weights are coefficients of a univariate
Chebyshev polynomial obtained by solving a sum-of-squares optimization problem
in the offline step. Hence, given a finite number of moments of probability
distributions, risk can be estimated in real-time. We demonstrate the
performance of the provided approach by solving probabilistic collision
checking problems where we aim to find the probability of collision of a robot
with a non-convex obstacle in the presence of probabilistic uncertainties in
the location of the robot and size, location, and geometry of the obstacle.Comment: 57th IEEE Conference on Decision and Control 201
Real-Time Tube-Based Non-Gaussian Risk Bounded Motion Planning for Stochastic Nonlinear Systems in Uncertain Environments via Motion Primitives
We consider the motion planning problem for stochastic nonlinear systems in
uncertain environments. More precisely, in this problem the robot has
stochastic nonlinear dynamics and uncertain initial locations, and the
environment contains multiple dynamic uncertain obstacles. Obstacles can be of
arbitrary shape, can deform, and can move. All uncertainties do not necessarily
have Gaussian distribution. This general setting has been considered and solved
in [1]. In addition to the assumptions above, in this paper, we consider
long-term tasks, where the planning method in [1] would fail, as the
uncertainty of the system states grows too large over a long time horizon.
Unlike [1], we present a real-time online motion planning algorithm. We build
discrete-time motion primitives and their corresponding continuous-time tubes
offline, so that almost all system states of each motion primitive are
guaranteed to stay inside the corresponding tube. We convert probabilistic
safety constraints into a set of deterministic constraints called risk
contours. During online execution, we verify the safety of the tubes against
deterministic risk contours using sum-of-squares (SOS) programming. The
provided SOS-based method verifies the safety of the tube in the presence of
uncertain obstacles without the need for uncertainty samples and time
discretization in real-time. By bounding the probability the system states
staying inside the tube and bounding the probability of the tube colliding with
obstacles, our approach guarantees bounded probability of system states
colliding with obstacles. We demonstrate our approach on several long-term
robotics tasks.Comment: International Conference on Intelligent Robots and Systems (IROS),
202
Convex Risk Bounded Continuous-Time Trajectory Planning and Tube Design in Uncertain Nonconvex Environments
In this paper, we address the trajectory planning problem in uncertain
nonconvex static and dynamic environments that contain obstacles with
probabilistic location, size, and geometry. To address this problem, we provide
a risk bounded trajectory planning method that looks for continuous-time
trajectories with guaranteed bounded risk over the planning time horizon. Risk
is defined as the probability of collision with uncertain obstacles. Existing
approaches to address risk bounded trajectory planning problems either are
limited to Gaussian uncertainties and convex obstacles or rely on
sampling-based methods that need uncertainty samples and time discretization.
To address the risk bounded trajectory planning problem, we leverage the notion
of risk contours to transform the risk bounded planning problem into a
deterministic optimization problem. Risk contours are the set of all points in
the uncertain environment with guaranteed bounded risk. The obtained
deterministic optimization is, in general, nonlinear and nonconvex time-varying
optimization. We provide convex methods based on sum-of-squares optimization to
efficiently solve the obtained nonconvex time-varying optimization problem and
obtain the continuous-time risk bounded trajectories without time
discretization. The provided approach deals with arbitrary (and known)
probabilistic uncertainties, nonconvex and nonlinear, static and dynamic
obstacles, and is suitable for online trajectory planning problems. In
addition, we provide convex methods based on sum-of-squares optimization to
build the max-sized tube with respect to its parameterization along the
trajectory so that any state inside the tube is guaranteed to have bounded
risk.Comment: Accepted by IJRR (extension of RSS 2021 paper arXiv:2106.05489
invited to IJRR
Non-Gaussian Uncertainty Minimization Based Control of Stochastic Nonlinear Robotic Systems
In this paper, we consider the closed-loop control problem of nonlinear
robotic systems in the presence of probabilistic uncertainties and
disturbances. More precisely, we design a state feedback controller that
minimizes deviations of the states of the system from the nominal state
trajectories due to uncertainties and disturbances. Existing approaches to
address the control problem of probabilistic systems are limited to particular
classes of uncertainties and systems such as Gaussian uncertainties and
processes and linearized systems. We present an approach that deals with
nonlinear dynamics models and arbitrary known probabilistic uncertainties. We
formulate the controller design problem as an optimization problem in terms of
statistics of the probability distributions including moments and
characteristic functions. In particular, in the provided optimization problem,
we use moments and characteristic functions to propagate uncertainties
throughout the nonlinear motion model of robotic systems. In order to reduce
the tracking deviations, we minimize the uncertainty of the probabilistic
states around the nominal trajectory by minimizing the trace and the
determinant of the covariance matrix of the probabilistic states. To obtain the
state feedback gains, we solve deterministic optimization problems in terms of
moments, characteristic functions, and state feedback gains using off-the-shelf
interior-point optimization solvers. To illustrate the performance of the
proposed method, we compare our method with existing probabilistic control
methods.Comment: International Conference on Intelligent Robots and Systems (IROS),
202