Autonomous navigation through unknown environments is a challenging task that entails real-time localization, perception, planning, and control. UAVs with this capability have begun to emerge in the literature with advances in lightweight sensing and computing. Although the planning methodologies vary from platform to platform, many algorithms adopt a hierarchical planning architecture where a slow, low-fidelity global planner guides a fast, high-fidelity local planner. However, in unknown environments, this approach can lead to erratic or unstable behavior due to the interaction between the global planner, whose solution is changing constantly, and the local planner; a consequence of not capturing higher-order dynamics in the global plan. This work proposes a planning framework in which multi-fidelity models are used to reduce the discrepancy between the local and global planner. Our approach uses high-, medium-, and low-fidelity models to compose a path that captures higher-order dynamics while remaining computationally tractable. In addition, we address the interaction between a fast planner and a slower mapper by considering the sensor data not yet fused into the map during the collision check. This novel mapping and planning framework for agile flights is validated in simulation and hardware experiments, showing replanning times of 5-40 ms in cluttered environments.
For , most of the current state-of-the-art methods exploit the differential flatness of the quadrotors, and solve the planning problem minimizing the squared norm of a derivative of the position @cite_13 , @cite_21 to find a dynamically feasible smooth trajectory @cite_12 . On the one hand, there are approaches where the obstacle constraints, and sometimes also input constraints, are checked after solving the optimization problem: Some of them use stitched polynomial trajectories that pass through several waypoints obtained running RRT-based methods @cite_13 , @cite_12 , @cite_16 . Others use Pontryagin's Minimum Principle to find closed-form solutions @cite_3 . These closed-form solutions are also used to search over the state space @cite_17 , @cite_18 . Alternatively, there are works that use the cost function to penalize the distance to the obstacles @cite_24 , @cite_11 , requiring usually computationally expensive distance fields representations. Finally, there are also approaches that add the obstacle constraints in the optimization problem, like Mixed-Integer Programming (MIP) @cite_22 , convex decompositions @cite_20 , @cite_15 , @cite_5 and Successive Convexification @cite_6 , @cite_4 , @cite_8 .
@cite_18 @cite_11 @cite_4 @cite_22 @cite_8 @cite_21 @cite_20 @cite_3 @cite_6 @cite_24 @cite_5 @cite_15 @cite_16 @cite_13 @cite_12 @cite_17
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We explore the challenges of planning trajectories for quadrotors through cluttered indoor environments. We extend the existing work on polynomial trajectory generation by presenting a method of jointly optimizing polynomial path segments in an unconstrained quadratic program that is numerically stable for high-order polynomials and large numbers of segments, and is easily formulated for efficient sparse computation. We also present a technique for automatically selecting the amount of time allocated to each segment, and hence the quadrotor speeds along the path, as a function of a single parameter determining aggressiveness, subject to actuator constraints. The use of polynomial trajectories, coupled with the differentially flat representation of the quadrotor, eliminates the need for computationally intensive sampling and simulation in the high dimensional state space of the vehicle during motion planning. Our approach generates high-quality trajecrtories much faster than purely sampling-based optimal kinodynamic planning methods, but sacrifices the guarantee of asymptotic convergence to the global optimum that those methods provide. We demonstrate the performance of our algorithm by efficiently generating trajectories through challenging indoor spaces and successfully traversing them at speeds up to 8 m s. A demonstration of our algorithm and flight performance is available at: http: groups.csail.mit.edu rrg quad_polynomial_trajectory_planning. In this paper, we study decomposition methods based on separable approximations for minimizing the augmented Lagrangian. In particular, we study and compare the diagonal quadratic approximation method DQAM of Mulvey and Ruszczynski [A diagonal quadratic approximation method for large scale linear programs, Oper. Res. Lett. 12 1992, pp. 205–215] and the parallel coordinate descent method PCDM of Richtarik and Takac [Parallel coordinate descent methods for big data optimization. Technical report, November 2012. arXiv:1212.0873]. We show that the two methods are equivalent for feasibility problems up to the selection of a step-size parameter. Furthermore, we prove an improved complexity bound for PCDM under strong convexity, and show that this bound is at least 8L′ Lω−12 times better than the best known bound for DQAM, where ω is the degree of partial separability and L’ and L are the maximum and average of the block Lipschitz constants of the gradient of the quadratic penalty appearing in the augmented Lagrangian. We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger. We present a new approach to the design of smooth trajectories for quadrotor unmanned aerial vehicles (UAVs), which are free of collisions with obstacles along their entire length. To avoid the non-convex constraints normally required for obstacle-avoidance, we perform a mixed-integer optimization in which polynomial trajectories are assigned to convex regions which are known to be obstacle-free. Prior approaches have used the faces of the obstacles themselves to define these convex regions. We instead use IRIS, a recently developed technique for greedy convex segmentation [1], to pre-compute convex regions of safe space. This results in a substantially reduced number of integer variables, which improves the speed with which the optimization can be solved to its global optimum, even for tens or hundreds of obstacle faces. In addition, prior approaches have typically enforced obstacle avoidance at a finite set of sample or knot points. We introduce a technique based on sums-of-squares (SOS) programming that allows us to ensure that the entire piecewise polynomial trajectory is free of collisions using convex constraints. We demonstrate this technique in 2D and in 3D using a dynamical model in the Drake toolbox for Matlab [2]. The thesis describes a new method for obtaining minimizers for optimal control problems whose minima serve as control policies for guiding nonlinear dynamical systems to achieve prescribed goals under imposed trajectory and actuator constraints. One of the major contributions of the present work resides in the approximation of such minimizers by piecewise polynomial functions expressed in terms of a linear combination of non-uniform rational B-spline (NURBS) basis functions and the judicious exploitation of the properties of the resulting NURBS curves to improve the computational effort often associated with solving optimal control problems for constrained dynamical systems. In particular, by exploiting the two structures combined in a NURBS curve, NURBS basis functions and an associated union of overlapping polytopes constructed from the coefficients of the linear combination, we are able to separate an optimal control problem into two subproblems | guidance and obstacle avoidance, making the original problem tractable. This is accomplished by laying out the union of overlapping polytopes in such a way that they delineate a section of space that avoids all obstacles and then manipulating the NURBS basis functions to obtain trajectories that are guaranteed to remain bounded by this section of space without explicitly including the conjunction of disjunctions naturally induced from obstacles into the guidance problem. In addition, we show how one can construct systematically a feasible region that corresponds to a NURBS parameterization starting from an ordered union of pairwise adjacently overlapping nonempty compact convex sets. Specifically, we show how to setup a nonlinear programming problem to solve for the feasible region in terms of an ordered union of pairwise adjacently overlapping polytopes with nonempty interiors by maximizing the sum of their volumes and starting from a feasible region described by an ordered union of pairwise adjacently overlapping nonempty convex compact simi-algebraic sets. Finally, we show how this strategy can be implemented practically for an autonomous system traversing an urban environment. Finally, this work culminated in the filing of patent US20070179685 on behalf of Northrop Grumman for the Space Technology sector and in the development of the NURBS-based OTG software package. This C++ package contains the theoretical results of this thesis in the form of an object-oriented implementation optimized for real-time trajectory generation. In this work, we propose a search-based planning method to compute dynamically feasible trajectories for a quadrotor flying in an obstacle-cluttered environment. Our approach searches for smooth, minimum-time trajectories by exploring the map using a set of short-duration motion primitives. The primitives are generated by solving an optimal control problem and induce a finite lattice discretization on the state space which can be explored using a graph-search algorithm. The proposed approach is able to generate resolution-complete (i.e., optimal in the discretized space), safe, dynamically feasibility trajectories efficiently by exploiting the explicit solution of a Linear Quadratic Minimum Time problem. It does not assume a hovering initial condition and, hence, is suitable for fast online re-planning while the robot is moving. Quadrotor navigation with online re-planning is demonstrated using the proposed approach in simulation and physical experiments and comparisons with trajectory generation based on state-of-art quadratic programming are presented. We focus on a replanning scenario for quadrotors where considering time efficiency, non-static initial state and dynamical feasibility is of great significance. We propose a real-time B-spline based kinodynamic (RBK) search algorithm, which transforms a position-only shortest path search (such as A * and Dijkstra) into an efficient kinodynamic search, by exploring the properties of B-spline parameterization. The RBK search is greedy and produces a dynamically feasible time-parameterized trajectory efficiently, which facilitates non-static initial state of the quadrotor. To cope with the limitation of the greedy search and the discretization induced by a grid structure, we adopt an elastic optimization (EO) approach as a post-optimization process, to refine the control point placement provided by the RBK search. The EO approach finds the optimal control point placement inside an expanded elastic tube which represents the free space, by solving a Quadratically Constrained Quadratic Programming (QCQP) problem. We design a receding horizon replanner based on the local control property of B-spline. A systematic comparison of our method against two state-of-the-art methods is provided. We integrate our replanning system with a monocular vision-based quadrotor and validate our performance onboard. There is extensive literature on using convex optimization to derive piece-wise polynomial trajectories for controlling differential flat systems with applications to three-dimensional flight for Micro Aerial Vehicles. In this work, we propose a method to formulate trajectory generation as a quadratic program (QP) using the concept of a Safe Flight Corridor (SFC). The SFC is a collection of convex overlapping polyhedra that models free space and provides a connected path from the robot to the goal position. We derive an efficient convex decomposition method that builds the SFC from a piece-wise linear skeleton obtained using a fast graph search technique. The SFC provides a set of linear inequality constraints in the QP allowing real-time motion planning. Because the range and field of view of the robot's sensors are limited, we develop a framework of Receding Horizon Planning , which plans trajectories within a finite footprint in the local map, continuously updating the trajectory through a re-planning process. The re-planning process takes between 50 to 300 ms for a large and cluttered map. We show the feasibility of our approach, its completeness and performance, with applications to high-speed flight in both simulated and physical experiments using quadrotors. We focus on nonconvex and nonsmooth minimization problems with a composite objective, where the differentiable part of the objective is freed from the usual and restrictive global Lipschitz gradient continuity assumption. This longstanding smoothness restriction is pervasive in first order methods (FOM), and was recently circumvent for convex composite optimization by Bauschke, Bolte and Teboulle, through a simple and elegant framework which captures, all at once, the geometry of the function and of the feasible set. Building on this work, we tackle genuine nonconvex problems. We first complement and extend their approach to derive a full extended descent lemma by introducing the notion of smooth adaptable functions. We then consider a Bregman-based proximal gradient methods for the nonconvex composite model with smooth adaptable functions, which is proven to globally converge to a critical point under natural assumptions on the problem's data. To illustrate the power and potential of our general framework and results, we consider a broad class of quadratic inverse problems with sparsity constraints which arises in many fundamental applications, and we apply our approach to derive new globally convergent schemes for this class. This paper investigates the computational complexity of planning the motion of a body B in 2-D or 3-D space, so as to avoid collision with moving obstacles of known, easily computed, trajectories. Dynamic movement problems are of fundamental importance to robotics, but their computational complexity has not previously been investigated. We provide evidence that the 3-D dynamic movement problem is intractable even if B has only a constant number of degrees of freedom of movement. In particular, we prove the problem is PSPACE-hard if B is given a velocity modulus bound on its movements and is NP-hard even if B has no velocity modulus bound, where, in both cases, B has 6 degrees of freedom. To prove these results, we use a unique method of simulation of a Turing machine that uses time to encode configurations (whereas previous lower bound proofs in robotic motion planning used the system position to encode configurations and so required unbounded number of degrees of freedom). We also investigate a natural class of dynamic problems that we call asteroid avoidance problems : B, the object we wish to move, is a convex polyhedron that is free to move by translation with bounded velocity modulus, and the polyhedral obstacles have known translational trajectories but cannot rotate. This problem has many applications to robot, automobile, and aircraft collision avoidance. Our main positive results are polynomial time algorithms for the 2-D asteroid avoidance problem, where B is a moving polygon and we assume a constant number of obstacles, as well as single exponential time or polynomial space algorithms for the 3-D asteroid avoidance problem, where B is a convex polyhedron and there are arbitrarily many obstacles. Our techniques for solving these asteroid avoidance problems use “normal path” arguments, which are an intereting generalization of techniques previously used to solve static shortest path problems. We also give some additional positive results for various other dynamic movers problems, and in particular give polynomial time algorithms for the case in which B has no velocity bounds and the movements of obstacles are algebraic in space-time. Minimization with orthogonality constraints (e.g., (X^ X = I )) and or spherical constraints (e.g., ( x _2 = 1 )) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. To deal with these difficulties, we apply the Cayley transform—a Crank-Nicolson-like update scheme—to preserve the constraints and based on it, develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics. The efficiency of the proposed algorithms is demonstrated on a variety of test problems. In particular, for the maxcut problem, it exactly solves a decomposition formulation for the SDP relaxation. For polynomial optimization, nearest correlation matrix estimation and extreme eigenvalue problems, the proposed algorithms run very fast and return solutions no worse than those from their state-of-the-art algorithms. For the quadratic assignment problem, a gap 0.842 to the best known solution on the largest problem “tai256c” in QAPLIB can be reached in 5 min on a typical laptop. This research deals with the computation of optimal trajectories considering state and input constraints for linear and nonlinear systems that admit a polynomial representation through differential flatness. Based on a polynomial spline parameterization of the flat output an optimization problem in terms of the B-spline coefficients is derived that guarantees constraint satisfaction over the entire time horizon whereas classical approaches in the literature only impose the constraints on a finite time grid. As the proposed constraints are only sufficient conditions, a novel method is presented that effectively reduces their conservatism. Two numerical examples, a linear benchmark tracking problem and an optimal quadrotor maneuver, illustrate the efficiency and practicality of the presented method. In this paper we first study a smooth optimization approach for solving a class of nonsmooth strictly concave maximization problems whose objective functions admit smooth convex minimization reformulations. In particular, we apply Nesterov's smooth optimization technique [Y. E. Nesterov, Dokl. Akad. Nauk SSSR, 269 (1983), pp. 543-547; Y. E. Nesterov, Math. Programming, 103 (2005), pp. 127-152] to their dual counterparts that are smooth convex problems. It is shown that the resulting approach has @math iteration complexity for finding an @math -optimal solution to both primal and dual problems. We then discuss the application of this approach to sparse covariance selection that is approximately solved as an @math -norm penalized maximum likelihood estimation problem, and also propose a variant of this approach which has substantially outperformed the latter one in our computational experiments. We finally compare the performance of these approaches with other first-order methods, namely, Nesterov's @math smooth approximation scheme and block-coordinate descent method studied in [A. d'Aspremont, O. Banerjee, and L. El Ghaoui, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 56-66; J. Friedman, T. Hastie, and R. Tibshirani, Biostatistics, 9 (2008), pp. 432-441] for sparse covariance selection on a set of randomly generated instances. It shows that our smooth optimization approach substantially outperforms the first method above, and moreover, its variant substantially outperforms both methods above. This paper examines techniques for finding falsifying trajectories of hybrid systems using an approach that we call trajectory splicing. Many formal verification techniques for hybrid systems, including flowpipe construction, can identify plausible abstract counterexamples for property violations. However, there is often a gap between the reported abstract counterexamples and the concrete system trajectories. Our approach starts with a candidate sequence of disconnected trajectory segments, each segment lying inside a discrete mode. However, such disconnected segments do not form concrete violations due to the gaps that exist between the ending state of one segment and the starting state of the subsequent segment. Therefore, trajectory splicing uses local optimization to minimize the gap between these segments, effectively splicing them together to form a concrete trajectory. We demonstrate the use of our approach for falsifying safety properties of hybrid systems using standard optimization techniques. As such, our approach is not restricted to linear systems. We compare our approach with other falsification approaches including uniform random sampling and a robustness guided falsification approach used in the tool S-Taliro. Our preliminary evaluation clearly shows the potential of our approach to search for candidate trajectory segments and use them to find concrete property violations. Direct methods for trajectory optimization are widely used for planning locally optimal trajectories of robotic systems. Many critical tasks, such as locomotion and manipulation, often involve impacting the ground or objects in the environment. Most state-of-the-art techniques treat the discontinuous dynamics that result from impacts as discrete modes and restrict the search for a complete path to a specified sequence through these modes. Here we present a novel method for trajectory planning of rigid-body systems that contact their environment through inelastic impacts and Coulomb friction. This method eliminates the requirement for a priori mode ordering. Motivated by the formulation of multi-contact dynamics as a Linear Complementarity Problem for forward simulation, the proposed algorithm poses the optimization problem as a Mathematical Program with Complementarity Constraints. We leverage Sequential Quadratic Programming to naturally resolve contact constraint forces while simultaneously optimizing a trajectory that satisfies the complementarity constraints. The method scales well to high-dimensional systems with large numbers of possible modes. We demonstrate the approach on four increasingly complex systems: rotating a pinned object with a finger, simple grasping and manipulation, planar walking with the Spring Flamingo robot, and high-speed bipedal running on the FastRunner platform. These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions.