Complete analytic solution to Brownian unicycle dynamics

This paper derives a complete analytical solution for the probability distribution of the configuration of a non-holonomic vehicle that moves in two spatial dimensions by satisfying the unicycle kinematic constraints and in presence of Brownian noises. In contrast to previous solutions, the one here derived holds even in the case of arbitrary linear and angular speed. This solution is obtained by deriving the analytical expression of any-order moment of the probability distribution. To the best of our knowledge, an analytical expression for any-order moment that holds even in the case of arbitrary linear and angular speed, has never been derived before. To compute these moments, a direct integration of the Langevin equation is carried out and each moment is expressed as a multiple integral of the deterministic motion (i.e., the known motion that would result in absence of noise). For the special case when the ratio between the linear and angular speed is constant, the multiple integrals can be easily solved and expressed as the real or the imaginary part of suitable analytic functions. As an application of the derived analytical results, the paper investigates the diffusivity of the considered Brownian motion for constant and for arbitrary time-dependent linear and angular speed.Comment: 22 pages, 6 figures, 2 table

Feedback-based Information Roadmap (FIRM): Graph-based Estimation and Control of Robotic Systems Under Uncertainty

This dissertation addresses the problem of stochastic optimal control with imperfect measurements. The main application of interest is robot motion planning under uncertainty. In the presence of process uncertainty and imperfect measurements, the system's state is unknown and a state estimation module is required to provide the information-state (belief), which is the probability distribution function (pdf) over all possible states. Accordingly, successful robot operation in such a setting requires reasoning about the evolution of information-state and its quality in future time steps. In its most general form, this is modeled as a Partially-Observable Markov Decision Process (POMDP) problem. Unfortunately, however, the exact solution of this problem over continuous spaces in the presence of constraints is computationally intractable. Correspondingly, state-of-the-art methods that provide approximate solutions are limited to problems with short horizons and small domains. The main challenge for these problems is the exponential growth of the search tree in the information space, as well as the dependency of the entire search tree on the initial belief. Inspired by sampling-based (roadmap-based) methods, this dissertation proposes a method to construct a "graph" in information space, called Feedback-based Information RoadMap (FIRM). Each FIRM node is a probability distribution and each FIRM edge is a local controller. The concept of belief stabilizers is introduced as a way to steer the current belief toward FIRM nodes and induce belief reachability. The solution provided by the FIRM framework is a feedback law over the information space, which is obtained by switching among locally distributed feedback controllers. Exploiting such a graph in planning, the intractable POMDP problem over continuous spaces is reduced to a tractable MDP (Markov Decision Process) problem over the graph (FIRM) nodes. FIRM is the first graph generated in the information space that preserves the principle of optimality, i.e., the costs associated with different edges of FIRM are independent of each other. Unlike the forward search methods on tree-structures, the plans produced by FIRM are independent of the initial belief (i.e., plans are query-independent). As a result, they are robust and reliable. They are robust in the sense that if the system's belief deviates from the planned belief, then replanning is feasible in real-time, as the computed solution is a feedback over the entire belief graph. Computed plans are reliable in the sense that the probability of violating constraints (e.g., hitting obstacles) can be seamlessly incorporated into the planning law. Moreover, FIRM is a scalable framework, as the computational complexity of its construction is linear in the size of underlying graph as opposed to state-of-the-art methods whose complexity is exponential in the size of underlying graph. In addition to the abstract framework, we present concrete FIRM instantiations for three main classes of robotic systems: holonomic, nonholonomic, and non-pointstabilizable. The abstract framework opens new avenues for extending FIRM to a broader class of systems that are not considered in this dissertation. This includes systems with discrete dynamics or in general systems that are not well-linearizable, systems with non-Gaussian distributions, and systems with unobservable modes. In addition to the abstract framework and concrete instantiations of it, we propose a formal technique for replanning with FIRM based on a rollout-policy algorithm to handle changes in the environment as well as discrepancies between actual and computational models. We demonstrate the performance of the proposed motion planning method on different robotic systems, both in simulation and on physical systems. In the problems we consider, the system is subject to motion and sensing noise. Our results demonstrate a significant advance over existing approaches for motion planning in information space. We believe the proposed framework takes an important step toward making information space planners applicable to real world robotic applications

Complete analytic solution to Brownian unicycle dynamics

This paper derives a complete analytical solution for the probability distribution of the configuration of a non-holonomic mobile robot that moves in two spatial dimensions by satisfying the unicycle kinematic constraints. The proposed solution differs from previous solutions since it is obtained by deriving the analytical expression of any-order moment of the probability distribution. To the best of our knowledge, an analytical expression for any-order moment that holds even in the case of arbitrary linear and angular speed, has never been derived before. To compute these moments, a direct integration of the Langevin equation is carried out and each moment is expressed as a multiple integral of the deterministic motion (i.e., the known motion that would result in absence of noise). For the special case when the ratio between the linear and angular speed is constant, the multiple integrals can be easily solved and expressed as the real or the imaginary part of suitable analytic functions. As an application of the derived analytical results, the paper investigates the diffusivity of the considered Brownian motion for constant and for arbitrary time-dependent linear and angular speed

Generalized Sampling-Based Feedback Motion Planners

The motion planning problem can be formulated as a Markov decision process (MDP), if the uncertainties in the robot motion and environments can be modeled probabilistically. The complexity of solving these MDPs grow exponentially as the dimension of the problem increases and hence, it is nearly impossible to solve the problem even without constraints. Using hierarchical methods, these MDPs can be transformed into a semi-Markov decision process (SMDP) which only needs to be solved at certain landmark states. In the deterministic robotics motion planning community, sampling based algorithms like probabilistic roadmaps (PRM) and rapidly exploring random trees (RRTs) have been successful in solving very high dimensional deterministic problem. However they are not robust to system with uncertainties in the system dynamics and hence, one of the primary objective of this work is to generalize PRM/RRT to solve motion planning with uncertainty. We first present generalizations of randomized sampling based algorithms PRM and RRT, to incorporate the process uncertainty, and obstacle location uncertainty, termed as "generalized PRM" (GPRM) and "generalized RRT" (GRRT). The controllers used at the lower level of these planners are feedback controllers which ensure convergence of trajectories while mitigating the effects of process uncertainty. The results indicate that the algorithms solve the motion planning problem for a single agent in continuous state/control spaces in the presence of process uncertainty, and constraints such as obstacles and other state/input constraints. Secondly, a novel adaptive sampling technique, termed as "adaptive GPRM" (AGPRM), is proposed for these generalized planners to increase the efficiency and overall success probability of these planners. It was implemented on high-dimensional robot n-link manipulators, with up to 8 links, i.e. in a 16-dimensional state-space. The results demonstrate the ability of the proposed algorithm to handle the motion planning problem for highly non-linear systems in very high-dimensional state space. Finally, a solution methodology, termed the "multi-agent AGPRM" (MAGPRM), is proposed to solve the multi-agent motion planning problem under uncertainty. The technique uses a existing solution technique to the multiple traveling salesman problem (MTSP) in conjunction with GPRM. For real-time implementation, an ?inter-agent collision detection and avoidance? module was designed which ensures that no two agents collide at any time-step. Algorithm was tested on teams of homogeneous and heterogeneous agents in cluttered obstacle space and the algorithm demonstrate the ability to handle such problems in continuous state/control spaces in presence of process uncertainty

Proprioceptive Invariant Robot State Estimation

This paper reports on developing a real-time invariant proprioceptive robot state estimation framework called DRIFT. A didactic introduction to invariant Kalman filtering is provided to make this cutting-edge symmetry-preserving approach accessible to a broader range of robotics applications. Furthermore, this work dives into the development of a proprioceptive state estimation framework for dead reckoning that only consumes data from an onboard inertial measurement unit and kinematics of the robot, with two optional modules, a contact estimator and a gyro filter for low-cost robots, enabling a significant capability on a variety of robotics platforms to track the robot's state over long trajectories in the absence of perceptual data. Extensive real-world experiments using a legged robot, an indoor wheeled robot, a field robot, and a full-size vehicle, as well as simulation results with a marine robot, are provided to understand the limits of DRIFT

On-line learning and updating unmanned tracked vehicle dynamics

Increasing levels of autonomy impose more pronounced performance requirements for unmanned ground vehicles (UGV). Presence of model uncertainties significantly reduces a ground vehicle performance when the vehicle is traversing an unknown terrain or the vehicle inertial parameters vary due to a mission schedule or external disturbances. A comprehensive mathematical model of a skid steering tracked vehicle is presented in this paper and used to design a control law. Analysis of the controller under model uncertainties in inertial parameters and in the vehicle-terrain interaction revealed undesirable behavior, such as controller divergence and offset from the desired trajectory. A compound identification scheme utilizing an exponential forgetting recursive least square, generalized Newton–Raphson (NR), and Unscented Kalman Filter methods is proposed to estimate the model parameters, such as the vehicle mass and inertia, as well as parameters of the vehicle-terrain interaction, such as slip, resistance coefficients, cohesion, and shear deformation modulus on-line. The proposed identification scheme facilitates adaptive capability for the control system, improves tracking performance and contributes to an adaptive path and trajectory planning framework, which is essential for future autonomous ground vehicle mission

Design of an UAV swarm

This master thesis tries to give an overview on the general aspects involved in the design of an UAV swarm. UAV swarms are continuoulsy gaining popularity amongst researchers and UAV manufacturers, since they allow greater success rates in task accomplishing with reduced times. Appart from this, multiple UAVs cooperating between them opens a new field of missions that can only be carried in this way. All the topics explained within this master thesis will explain all the agents involved in the design of an UAV swarm, from the communication protocols between them, navigation and trajectory analysis and task allocation

Master of Science

thesisThis thesis details the development of the Algorithmic Robotics Laboratory, its experimental software environment, and a case study featuring a novel hardware validation of optimal reciprocal collision avoidance. We constructed a robotics laboratory in both software and hardware in which to perform our experiments. This lab features a netted flying volume with motion capture and two custom quadrotors. Also, two experimental software architectures are developed for actuating both ground and aerial robots within a Linux Robot Operating System environment. The first of the frameworks is based upon a single finite state machine program which managed each aspect of the experiment. Concerns about the complexity and reconfigurability of the finite state machine prompted the development of a second framework. This final framework is a multimodal structure featuring programs which focus on these specific functions: State Estimation, Robot Drivers, Experimental Controllers, Inputs, Human Robot Interaction, and a program tailored to the specifics of the algorithm tested in the experiment. These modular frameworks were used to fulfill the mission of the Algorithmic Robotics Lab, in that they were developed to validate robotics algorithms in experiments that were previously only shown in simulation. A case study into collision avoidance was used to mark the foundation of the laboratory through the proving of an optimal reciprocal collision avoidance algorithm for the first time in hardware. In the case study, two human-controlled quadrotors were maliciously flown in colliding trajectories. Optimal reciprocal collision avoidance was demonstrated for the first time on completely independent agents with local sensing. The algorithm was shown to be robust to violations of its inherent assumptions about the dynamics of agents and the ability for those agents to sense imminent collisions. These experiments, in addition to the mathematical foundation of exponential convergence, submits th a t optimal reciprocal collision avoidance is a viable method for holonomic robots in both 2-D and 3-D with noisy sensing. A basis for the idea of reciprocal dance, a motion often seen in human collision avoidance, is also suggested in demonstration to be a product of uncertainty about the state of incoming agents. In the more than one hundred tests conducted in multiple environments, no midair collisions were ever produced