Controller Synthesis for Discrete-Time Polynomial Systems via Occupation Measures

In this paper, we design nonlinear state feedback controllers for discrete-time polynomial dynamical systems via the occupation measure approach. We propose the discrete-time controlled Liouville equation, and use it to formulate the controller synthesis problem as an infinite-dimensional linear programming problem on measures, which is then relaxed as finite-dimensional semidefinite programming problems on moments of measures and their duals on sums-of-squares polynomials. Nonlinear controllers can be extracted from the solutions to the relaxed problems. The advantage of the occupation measure approach is that we solve convex problems instead of generally non-convex problems, and the computational complexity is polynomial in the state and input dimensions, and hence the approach is more scalable. In addition, we show that the approach can be applied to over-approximating the backward reachable set of discrete-time autonomous polynomial systems and the controllable set of discrete-time polynomial systems under known state feedback control laws. We illustrate our approach on several dynamical systems

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

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

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

Involvement of Glutamate Transporter-1 in Neuroprotection against Global Brain Ischemia-Reperfusion Injury Induced by Postconditioning in Rats

Ischemic postconditioning refers to several transient reperfusion and ischemia cycles after an ischemic event and before a long duration of reperfusion. The procedure produces neuroprotective effects. The mechanisms underlying these neuroprotective effects are poorly understood. In this study, we found that most neurons in the CA1 region died after 10 minutes of ischemia and is followed by 72 hours of reperfusion. However, brain ischemic postconditioning (six cycles of 10 s/10 s reperfusion/re-occlusion) significantly reduced neuronal death. Significant up-regulation of Glutamate transporter-1 was found after 3, 6, 24, 72 hours of reperfusion. The present study showed that ischemic postconditioning decreases cell death and that upregulation of GLT-1 expression may play an important role on this effect

Semidefinite programming approaches to multi-contact feedback control

This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2019Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 71-77).We consider the feedback design for stabilizing a rigid body system by making and breaking multiple contacts with the environment without prespecifying the timing or the number of occurrence of the contacts. We examine several different models of such systems and the roles of semidefinite programming and sums-of-squares programming in designing and verifying stabilizing controllers. First the system is modelled as a discrete-time piecewise affine system and we use semidefinite programming to design stabilizing controllers according to Lyapunov theory. Second the system is modelled as a discrete-time piecewise polynomial system and we use sums-of-squares programming to design feedback controllers. Third the system is modelled as a discrete-time polynomial system with linear complimentarity constraints for contacts and we use sums-of-squares to verify the controllers according to Lyapunov theory."Supported by MIT Cronin Fellowship, NASA Award NNX16AC49A, Air Force/Lincoln Laboratory Award No. 7000374874, Army Research Office Award No. W911NF-15-1-0166, and Department of the Navy, Office of Naval Research, Award No. N00014-18-1-2210"by Weiqiao Han.S.M.S.M. Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Scienc

Feedback design for multi-contact push recovery via LMI approximation of the Piecewise-Affine Quadratic Regulator

NASA (Award NNX16AC49A