159041459
Trajectory Optimization on Manifolds: A Theoretically-Guaranteed Embedded Sequential Convex Programming Approach
Sequential Convex Programming (SCP) has recently gained popularity as a tool for trajectory optimization due to its sound theoretical properties and practical performance. Yet, most SCP-based methods for trajectory optimization are restricted to Euclidean settings, which precludes their application to problem instances where one must reason about manifold-type constraints (that is, constraints, such as loop closure, which restrict the motion of a system to a subset of the ambient space). The aim of this paper is to fill this gap by extending SCP-based trajectory optimization methods to a manifold setting. The key insight is to leverage geometric embeddings to lift a manifold-constrained trajectory optimization problem into an equivalent problem defined over a space enjoying a Euclidean structure. This insight allows one to extend existing SCP methods to a manifold setting in a fairly natural way. In particular, we present a SCP algorithm for manifold problems with refined theoretical guarantees that resemble those derived for the Euclidean setting, and demonstrate its practical performance via numerical experiments.
582637
Fast marching tree: A fast marching sampling-based method for optimal motion planning in many dimensions
In this paper we present a novel probabilistic sampling-based motion planning algorithm called the Fast Marching Tree algorithm (FMT*). The algorithm is specifically aimed at solving complex motion planning problems in high-dimensional configuration spaces. This algorithm is proven to be asymptotically optimal and is shown to converge to an optimal solution faster than its state-of-the-art counterparts, chiefly PRM* and RRT*. The FMT* algorithm performs a ‘lazy’ dynamic programming recursion on a predetermined number of probabilistically drawn samples to grow a tree of paths, which moves steadily outward in cost-to-arrive space. As such, this algorithm combines features of both single-query algorithms (chiefly RRT) and multiple-query algorithms (chiefly PRM), and is reminiscent of the Fast Marching Method for the solution of Eikonal equations. As a departure from previous analysis approaches that are based on the notion of almost sure convergence, the FMT* algorithm is analyzed under the notion of convergence in probability: the extra mathematical flexibility of this approach allows for convergence rate bounds—the first in the field of optimal sampling-based motion planning. Specifically, for a certain selection of tuning parameters and configuration spaces, we obtain a convergence rate bound of order O(n −1/d+ρ ), where n is the number of sampled points, d is the dimension of the configuration space, and ρ is an arbitrarily small constant. We go on to demonstrate asymptotic optimality for a number of variations on FMT*, namely when the configuration space is sampled non-uniformly, when the cost is not arc length, and when connections are made based on the number of nearest neighbors instead of a fixed connection radius. Numerical experiments over a range of dimensions and obstacle configurations confirm our theoretical and heuristic arguments by showing that FMT*, for a given execution time, returns substantially better solutions than either PRM* or RRT*, especially in high-dimensional configuration spaces and in scenarios where collision-checking is expensive.
14876957
Sampling-based algorithms for optimal motion planning
During the last decade, sampling-based path planning algorithms, such as probabilistic roadmaps (PRM) and rapidly exploring random trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g. as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g. showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e. such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.
6327672
CHOMP: Gradient optimization techniques for efficient motion planning
Existing high-dimensional motion planning algorithms are simultaneously overpowered and underpowered. In domains sparsely populated by obstacles, the heuristics used by sampling-based planners to navigate “narrow passages” can be needlessly complex; furthermore, additional post-processing is required to remove the jerky or extraneous motions from the paths that such planners generate. In this paper, we present CHOMP, a novel method for continuous path refinement that uses covariant gradient techniques to improve the quality of sampled trajectories. Our optimization technique both optimizes higher-order dynamics and is able to converge over a wider range of input paths relative to previous path optimization strategies. In particular, we relax the collision-free feasibility prerequisite on input paths required by those strategies. As a result, CHOMP can be used as a standalone motion planner in many real-world planning queries. We demonstrate the effectiveness of our proposed method in manipulation planning for a 6-DOF robotic arm as well as in trajectory generation for a walking quadruped robot.
16670346
STOMP: Stochastic trajectory optimization for motion planning
We present a new approach to motion planning using a stochastic trajectory optimization framework. The approach relies on generating noisy trajectories to explore the space around an initial (possibly infeasible) trajectory, which are then combined to produced an updated trajectory with lower cost. A cost function based on a combination of obstacle and smoothness cost is optimized in each iteration. No gradient information is required for the particular optimization algorithm that we use and so general costs for which derivatives may not be available (e.g. costs corresponding to constraints and motor torques) can be included in the cost function. We demonstrate the approach both in simulation and on a mobile manipulation system for unconstrained and constrained tasks. We experimentally show that the stochastic nature of STOMP allows it to overcome local minima that gradient-based methods like CHOMP can get stuck in.
5579178
Funnel libraries for real-time robust feedback motion planning
We consider the problem of generating motion plans for a robot that are guaranteed to succeed despite uncertainty in the environment, parametric model uncertainty, and disturbances. Furthermore, we consider scenarios where these plans must be generated in real time, because constraints such as obstacles in the environment may not be known until they are perceived (with a noisy sensor) at runtime. Our approach is to pre-compute a library of “funnels” along different maneuvers of the system that the state is guaranteed to remain within (despite bounded disturbances) when the feedback controller corresponding to the maneuver is executed. We leverage powerful computational machinery from convex optimization (sums-of-squares programming in particular) to compute these funnels. The resulting funnel library is then used to sequentially compose motion plans at runtime while ensuring the safety of the robot. A major advantage of the work presented here is that by explicitly taking into account the effect of uncertainty, the robot can evaluate motion plans based on how vulnerable they are to disturbances. We demonstrate and validate our method using extensive hardware experiments on a small fixed-wing airplane avoiding obstacles at high speed (~12 mph), along with thorough simulation experiments of ground vehicle and quadrotor models navigating through cluttered environments. To our knowledge, these demonstrations constitute one of the first examples of provably safe and robust control for robotic systems with complex nonlinear dynamics that need to plan in real time in environments with complex geometric constraints.
51909372
Robust Tracking with Model Mismatch for Fast and Safe Planning: an SOS Optimization Approach
In the pursuit of real-time motion planning, a commonly adopted practice is to compute trajectories by running a planning algorithm on a simplified, low-dimensional dynamical model, and then employ a feedback tracking controller that tracks such a trajectory by accounting for the full, high-dimensional system dynamics. While this strategy of planning with model mismatch generally yields fast computation times, there are no guarantees of dynamic feasibility, which hampers application to safety-critical systems. Building upon recent work that addressed this problem through the lens of Hamilton-Jacobi (HJ) reachability, we devise an algorithmic framework whereby one computes, offline, for a pair of “planner” (i.e., low-dimensional) and “tracking” (i.e., high-dimensional) models, a feedback tracking controller and associated tracking bound. This bound is then used as a safety margin when generating motion plans via the low-dimensional model. Specifically, we harness the computational tool of sum-of-squares (SOS) programming to design a bilinear optimization algorithm for the computation of the feedback tracking controller and associated tracking bound. The algorithm is demonstrated via numerical experiments, with an emphasis on investigating the trade-off between the increased computational scalability afforded by SOS and its intrinsic conservativeness. Collectively, our results enable scaling the appealing strategy of planning with model mismatch to systems that are beyond the reach of HJ analysis, while maintaining safety guarantees.
53505960
GuSTO: Guaranteed Sequential Trajectory optimization via Sequential Convex Programming
Sequential Convex Programming (SCP) has recently seen a surge of interest as a tool for trajectory optimization. However, most available methods lack rigorous performance guarantees and they are often tailored to specific optimal control setups. In this paper, we present GuSTO (Guaranteed Sequential Trajectory optimization), an algorithmic framework to solve trajectory optimization problems for control-affine systems with drift. GuSTO generalizes earlier SCP-based methods for trajectory optimization (by addressing, for example, goal-set constraints and problems with either fixed or free final time) and enjoys theoretical convergence guarantees in terms of convergence to, at least, a stationary point. The theoretical analysis is further leveraged to devise an accelerated implementation of GuSTO, which originally infuses ideas from indirect optimal control into an SCP context. Numerical experiments on a variety of trajectory optimization setups show that GuSTO generally outperforms current state-of-the-art approaches in terms of success rates, solution quality, and computation times.
119879939
Solving Nonconvex Optimal Control Problems by Convex Optimization
Motivated by aerospace applications, this paper presents a methodology to use second-order cone programming to solve nonconvex optimal control problems. The nonconvexity arises from the presence of concave state inequality constraints and nonlinear terminal equality constraints. The development relies on a solution paradigm, in which the concave inequality constraints are approximated by successive linearization. Analysis is performed to establish the guaranteed satisfaction of the original inequality constraints, the existence of the successive solutions, and the equivalence of the solution of the original problem to the converged successive solution. These results lead to a rigorous proof of the convergence of the successive solutions under appropriate conditions as well as nonconservativeness of the converged solution. The nonlinear equality constraints are treated in a two-step procedure in which the constraints are first approximated by first-order expansions, then compensated by second-order correct...
1457699
Successive convexification of non-convex optimal control problems and its convergence properties
This paper presents an algorithm to solve non-convex optimal control problems, where non-convexity can arise from nonlinear dynamics, and non-convex state and control constraints. This paper assumes that the state and control constraints are already convex or convexified, the proposed algorithm convexifies the nonlinear dynamics, via a linearization, in a successive manner. Thus at each succession, a convex optimal control subproblem is solved. Since the dynamics are linearized and other constraints are convex, after a discretization, the subproblem can be expressed as a finite dimensional convex programming subproblem. Since convex optimization problems can be solved very efficiently, especially with custom solvers, this subproblem can be solved in time-critical applications, such as real-time path planning for autonomous vehicles. Several safe-guarding techniques are incorporated into the algorithm, namely virtual control and trust regions, which add another layer of algorithmic robustness. A convergence analysis is presented in continuous-time setting. By doing so, our convergence results will be independent from any numerical schemes used for discretization. Numerical simulations are performed for an illustrative trajectory optimization example.
893094
Motion planning with sequential convex optimization and convex collision checking
We present a new optimization-based approach for robotic motion planning among obstacles. Like CHOMP (Covariant Hamiltonian Optimization for Motion Planning), our algorithm can be used to find collision-free trajectories from naïve, straight-line initializations that might be in collision. At the core of our approach are (a) a sequential convex optimization procedure, which penalizes collisions with a hinge loss and increases the penalty coefficients in an outer loop as necessary, and (b) an efficient formulation of the no-collisions constraint that directly considers continuous-time safety Our algorithm is implemented in a software package called TrajOpt. We report results from a series of experiments comparing TrajOpt with CHOMP and randomized planners from OMPL, with regard to planning time and path quality. We consider motion planning for 7 DOF robot arms, 18 DOF full-body robots, statically stable walking motion for the 34 DOF Atlas humanoid robot, and physical experiments with the 18 DOF PR2. We also apply TrajOpt to plan curvature-constrained steerable needle trajectories in the SE(3) configuration space and multiple non-intersecting curved channels within 3D-printed implants for intracavitary brachytherapy. Details, videos, and source code are freely available at: http://rll.berkeley.edu/trajopt/ijrr.
4786809
Path Planning with Loop Closure Constraints Using an Atlas-Based RRT
In many relevant path planning problems, loop closure constraints reduce the configuration space to a manifold embedded in the higher-dimensional joint ambient space. Whereas many progresses have been done to solve path planning problems in the presence of obstacles, only few work consider loop closure constraints. In this paper we present the AtlasRRT algorithm, a planner specially tailored for such constrained systems that builds on recently developed tools for higher-dimensional continuation. These tools provide procedures to define charts that locally parametrize manifolds and to coordinate them forming an atlas. AtlasRRT simultaneously builds an atlas and a Rapidly-Exploring Random Tree (RRT), using the atlas to sample relevant configurations for the RRT, and the RRT to devise directions of expansion for the atlas. The new planner is advantageous since samples obtained from the atlas allow a more efficient extension of the RRT than state of the art approaches, where samples are generated in the joint ambient space.
12307449
Generation of collision-free trajectories for a quadrocopter fleet: A sequential convex programming approach
This paper presents an algorithm that generates collision-free trajectories in three dimensions for multiple vehicles within seconds. The problem is cast as a non-convex optimization problem, which is iteratively solved using sequential convex programming that approximates non-convex constraints by using convex ones. The method generates trajectories that account for simple dynamics constraints and is thus independent of the vehicle's type. An extensive a posteriori vehicle-specific feasibility check is included in the algorithm. The algorithm is applied to a quadrocopter fleet. Experimental results are shown.
1597660
Trajectory generation and control for precise aggressive maneuvers with quadrotors
We study the problem of designing dynamically feasible trajectories and controllers that drive a quadrotor to a desired state in state space. We focus on the development of a family of trajectories defined as a sequence of segments, each with a controller parameterized by a goal state or region in state space. Each controller is developed from the dynamic model of the robot and then iteratively refined through successive experimental trials in an automated fashion to account for errors in the dynamic model and noise in the actuators and sensors. We show that this approach permits the development of trajectories and controllers enabling such aggressive maneuvers as flying through narrow, vertical gaps and perching on inverted surfaces with high precision and repeatability.
14706601
Vision-Based State Estimation and Trajectory Control Towards High-Speed Flight with a Quadrotor
This paper addresses the development of a lightweight autonomous quadrotor that uses cameras and an inexpensive IMU as its only sensors and onboard processors for estimation and control. We describe a fully-functional, integrated system with a focus on robust visual-inertial state estimation, and demonstrate the quadrotor’s ability to autonomously travel at speeds up to 4 m/s and roll and pitch angles exceeding 20◦. The performance of the proposed system is demonstrated via challenging experiments in three dimensional indoor environments.
46954410
Trajectory Optimization On Manifolds with Applications to SO(3) and R3XS2
Manifolds are used in almost all robotics applications even if they are not explicitly modeled. We propose a differential geometric approach for optimizing trajectories on a Riemannian manifold with obstacles. The optimization problem depends on a metric and collision function specific to a manifold. We then propose our Safe Corridor on Manifolds (SCM) method of computationally optimizing trajectories for robotics applications via a constrained optimization problem. Our method does not need equality constraints, which eliminates the need to project back to a feasible manifold during optimization. We then demonstrate how this algorithm works on an example problem on SO(3) and a perception-aware planning example for visualinertially guided robots navigating in 3 dimensions. Formulating field of view constraints naturally results in modeling with the manifold R × S which cannot be modeled as a Lie group.
123365789
Feedback linearization of a flexible manipulator near its rigid body manifold
It is shown that a ‘slow’ 2n-dimensional manifold exists in the 4n-dimensional state space of an n-link manipulator with n flexible joints. While the flexible manifold is not linearizable by feedback, its restriction to'the slow manifold is. A method is given for computing a feedback control which achieves an approximate linearization.
108605633
A Mathematical Introduction to Robotic Manipulation
INTRODUCTION: Brief History. Multifingered Hands and Dextrous Manipulation. Outline of the Book. Bibliography. RIGID BODY MOTION: Rigid Body Transformations. Rotational Motion in R3. Rigid Motion in R3. Velocity of a Rigid Body. Wrenches and Reciprocal Screws. MANIPULATOR KINEMATICS: Introduction. Forward Kinematics. Inverse Kinematics. The Manipulator Jacobian. Redundant and Parallel Manipulators. ROBOT DYNAMICS AND CONTROL: Introduction. Lagrange's Equations. Dynamics of Open-Chain Manipulators. Lyapunov Stability Theory. Position Control and Trajectory Tracking. Control of Constrained Manipulators. MULTIFINGERED HAND KINEMATICS: Introduction to Grasping. Grasp Statics. Force-Closure. Grasp Planning. Grasp Constraints. Rolling Contact Kinematics. HAND DYNAMICS AND CONTROL: Lagrange's Equations with Constraints. Robot Hand Dynamics. Redundant and Nonmanipulable Robot Systems. Kinematics and Statics of Tendon Actuation. Control of Robot Hands. NONHOLONOMIC BEHAVIOR IN ROBOTIC SYSTEMS: Introduction. Controllability and Frobenius' Theorem. Examples of Nonholonomic Systems. Structure of Nonholonomic Systems. NONHOLONOMIC MOTION PLANNING: Introduction. Steering Model Control Systems Using Sinusoids. General Methods for Steering. Dynamic Finger Repositioning. FUTURE PROSPECTS: Robots in Hazardous Environments. Medical Applications for Multifingered Hands. Robots on a Small Scale: Microrobotics. APPENDICES: Lie Groups and Robot Kinematics. A Mathematica Package for Screw Calculus. Bibliography. Index Each chapter also includes a Summary, Bibliography, and Exercises
126392037
Task-Constrained Trajectory Planning of Free-Floating Space-Robotic Systems Using Convex Optimization
This paper addresses the trajectory-planning problem for a free-floating space-robotic system for which there are no actuations on the base spacecraft. The mission considered is that the end effect...
57759727
A convex-programming-based guidance algorithm to capture a tumbling object on orbit using a spacecraft equipped with a robotic manipulator
An algorithm to guide the capture of a tumbling resident space object by a spacecraft equipped with a robotic manipulator is presented. A solution to the guidance problem is found by solving a collection of convex programming problems. As convex programming offers deterministic convergence properties, this algorithm is suitable for onboard implementation and real-time use. A set of hardware-in-the-loop experiments substantiates this claim. To cast the guidance problem as a collection of convex programming problems, the capture maneuver is divided into two simultaneously occurring sub-maneuvers: a system-wide translation and an internal re-configuration. These two sub-maneuvers are optimized in two consecutive steps. A sequential convex programming procedure, overcoming the presence of non-convex constraints and nonlinear dynamics, is used on both optimization steps. A proof of convergence is offered for the system-wide translation, while a set of structured heuristics—trust regions—is used for the optimization of the internal re-configuration sub-maneuver. Videos of the numerically simulated and experimentally demonstrated maneuvers are included as supplementary material.
15967137
Tangent space RRT: A randomized planning algorithm on constraint manifolds
Motion planning for robots subject to holonomic constraints typically involves planning on constraint manifolds. In this paper we present the Tangent Space Rapidly Exploring Random Tree (TS-RRT) algorithm for planning on constraint manifolds. The key idea is to construct random trees not on the constraint manifold itself, but rather on tangent space approximations to the constraint manifold. Curvature-based methods are developed for constructing bounded tangent space approximations, as well as procedures for random node generation and bidirectional tree extension. Extensive numerical experiments suggest that the TS-RRT algorithm, despite its increased preprocessing and bookkeeping, outperforms existing constrained planning algorithms for a wide range of benchmark planning problems.
9310902
Smooth trajectory generation on SE(3) for a free flying space robot
We propose a new optimal trajectory generation technique on SE(3) which avoids known obstacles. We leverage techniques from differential geometry and Lie algebra to formulate a cost functional which is intrinsic to the geometric structure of this space and makes physical sense. We propose an approximation technique to generate trajectories on the subgroup SO(3) and use Semidefinite Programming (SDP) to approximate an NP-Hard problem with one which is tractable to compute. From this trajectory on the subgroup, the trajectory generation on the other dimensions of the group becomes a Quadratic Program (QP). For obstacle avoidance, we use a computational geometric technique to decompose the environment into overlapping convex regions to confine the trajectory. We show how this motion planning technique can be used to generate feasible trajectories for a space robot in SE(3) and describe controllers that enable the execution of the generated trajectory. We compare our method to other geometric techniques for calculating trajectories on SO(3) and SE(3), but in an obstacle-free environment.
5631216
Constrained motion planning for multiple vehicles on SE(3)
This paper proposes a computational method to solve constrained cooperative motion planning problems for multiple vehicles undergoing translational and rotational motions. The problem is solved by means of the Lie group projection operator approach, a recently developed optimization strategy for solving continuous-time optimal control problems on Lie groups. State constraints (for collision avoidance) are handled by means of a set of barrier functions, turning the optimization approach into an interior point method. A sample computation is shown to demonstrate the effectiveness of the method.
32405588
Optimal Control on Lie Groups: The Projection Operator Approach
Many nonlinear systems of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. Examples range from aircraft and underwater vehicles to quantum mechanical systems. In this paper, we develop an algorithm for solving continuous-time optimal control problems for systems evolving on (noncompact) Lie groups. This algorithm generalizes the projection operator approach for trajectory optimization originally developed for systems on vector spaces. Notions for generalizing system theoretic tools such as Riccati equations and linear and quadratic system approximations are developed. In this development, the covariant derivative of a map between two manifolds plays a key role in providing a chain rule for the required Lie group computations. An example optimal control problem on SO(3) is provided to highlight implementation details and to demonstrate the effectiveness of the method.
10430781
Reduction, Linearization, and Stability of Relative Equilibria for Mechanical Systems on Riemannian Manifolds
Abstract
Consider a Riemannian manifold equipped with an infinitesimal isometry. For this setup, a unified treatment is provided, solely in the language of Riemannian geometry, of techniques in reduction, linearization, and stability of relative equilibria. In particular, for mechanical control systems, an explicit characterization is given for the manner in which reduction by an infinitesimal isometry, and linearization along a controlled trajectory “commute.” As part of the development, relationships are derived between the Jacobi equation of geodesic variation and concepts from reduction theory, such as the curvature of the mechanical connection and the effective potential. As an application of our techniques, fiber and base stability of relative equilibria are studied. The paper also serves as a tutorial of Riemannian geometric methods applicable in the intersection of mechanics and control theory.

117041671
Introduction to Smooth Manifolds
Preface.- 1 Smooth Manifolds.- 2 Smooth Maps.- 3 Tangent Vectors.- 4 Submersions, Immersions, and Embeddings.- 5 Submanifolds.- 6 Sard's Theorem.- 7 Lie Groups.- 8 Vector Fields.- 9 Integral Curves and Flows.- 10 Vector Bundles.- 11 The Cotangent Bundle.- 12 Tensors.- 13 Riemannian Metrics.- 14 Differential Forms.- 15 Orientations.- 16 Integration on Manifolds.- 17 De Rham Cohomology.- 18 The de Rham Theorem.- 19 Distributions and Foliations.- 20 The Exponential Map.- 21 Quotient Manifolds.- 22 Symplectic Manifolds.- Appendix A: Review of Topology.- Appendix B: Review of Linear Algebra.- Appendix C: Review of Calculus.- Appendix D: Review of Differential Equations.- References.- Notation Index.- Subject Index
18215525
Control Theory from the Geometric Viewpoint
Geometrical methods have had a profound impact in the development of modern nonlinear control theory. Fundamental results such as the orbit theorem, feedback linearization, disturbance decoupling or the various controllability tests for nonlinear systems are all deeply rooted on a geometric view of control theory. It is perhaps surprising, and possibly debatable, that in order to understand and appreciate the “essence” of linear control systems one has to delve into the intricacies of Lie brackets and Lie algebras. This is because only the geometric perspective offers the tools to study the properties of control systems that are invariant under (nonlinear) changes of coordinates and can therefore be considered intrinsic. Consider, for example, an inverted pendulum or the ball and beam system. It should be apparent that reachability or optimality properties for these systems do not depend on the particular reference frame chosen to write their equations of motion. These are intrinsic properties of these physical systems and thus require geometric techniques for its study. “Control Theory from the Geometric Viewpoint” is a recent addition to the geometric control theory monograph/textbook literature having Jurdjevic (1997) as its closest neighbor and Nijmeijer and van der Schaft (1995), Isidori (1996) and Bloch (2003) as more distant ones. The book evolved from lecture notes for graduate courses taught by the first author at the International School for Advanced Studies in Trieste, Italy. The lecture notes style can be felt throughout the 24 chapters of the book treating a large number of topics ranging from controllability and reachability analysis to higher order conditions for optimality. This lecture notes style, patent on the relatively large number of treated topics in 400 pages, is the book’s main handicap and merit. If, on the one hand, most chapters can be independently read thus allowing the reader to immediately dive into the topic of choice and quickly reach the zenith result, on the other hand, there is a certain lack of fluidity when one tries to read the book chapters consecutively. In the remaining lines I will try to articulate my own opinion, naturally conditioned by my taste and background, on the choice of topics and presentation as I go through some of the individual chapters.
122758773
Maximum principle for optimal control problems with intermediate constraints
We consider optimal control problems with constraints at intermediate points of the trajectory. A natural technique (propagation of phase and control variables) is applied to reduce these problems to a standard optimal control problem of Pontryagin type with equality and inequality constraints at the trajectory endpoints. In this way we derive necessary optimality conditions that generalize the Pontryagin classical maximum principle. The same technique is applied to so-called variable structure problems and to some hybrid problems. The new optimality conditions are compared with the results of other authors and five examples illustrating their application are presented.
12187835
Jacobian linearisation in a geometric setting
Linearisation is a common technique in control applications, putting useful analysis and design methodologies at the disposal of the control engineer. In this paper, linearisation is studied from a differential geometric perspective. First it is pointed out that the "naive" Jacobian techniques do not make geometric sense along nontrivial reference trajectories, in that they are dependent on a choice of coordinates. A coordinate-invariant setting for linearisation is presented to address this matter. The setting here is somewhat more complicated than that seen in the naive setting. The controllability of the geometric linearisation is characterised by giving an alternate version of the usual controllability test for time-varying linear systems. The problems of stability, stabilisation, and quadratic optimal control are discussed as topics for future work.
125177510
Foundations of optimal control theory
Abstract : This complete and authoritative presentation of the current status of control theory offers a useful foundation for both study and research. With emphasis on general nonlinear differential systems, the book is carefully and systematically developed from elementary motivating examples, through the most comprehensive theory, to the final numerical solution of serious scientific and engineering control problems. The book features reviews of the most recent researches on processes described by partial differential equations, functional- differential, and delay-differential equations; the most recent treatment of impulse controllers, bounded rate controllers, feedback controllers, and bounded phase problems; and many unpublished new research results of the authors. In addition to an exhaustive treatment of the quantitative problems of optimal control, the qualitative concepts of stability, controllability, observability, and plant recognition receive a complete exposition. (Author)
123366068
Optimal control problems with delays in state and control variables subject to mixed control–state constraints
Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control–state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods are discussed by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and numerical examples from chemical engineering and economics illustrate the results. Copyright © 2008 John Wiley & Sons, Ltd.
7044532
Julia: A Fast Dynamic Language for Technical Computing
Dynamic languages have become popular for scientic computing. They are generally considered highly productive, but lacking in performance. This paper presents Julia, a new dynamic language for technical computing, designed for performance from the beginning by adapting and extending modern programming language techniques. A design based on generic functions and a rich type system simultaneously enables an expressive programming model and successful type inference, leading to good performance for a wide range of programs. This makes it possible for much of Julia’s library to be written in Julia itself, while also incorporating best-of-breed C and Fortran libraries.
58998497
Real-Time Collision Detection
Written by an expert in the game industry, Christer Ericson's new book is a comprehensive guide to the components of efficient real-time collision detection systems. The book provides the tools and know-how needed to implement industrial-strength collision detection for the highly detailed dynamic environments of applications such as 3D games, virtual reality applications, and physical simulators.Of the many topics covered, a key focus is on spatial and object partitioning through a wide variety of grids, trees, and sorting methods. The author also presents a large collection of intersection and distance tests for both simple and complex geometric shapes. Sections on vector and matrix algebra provide the background for advanced topics such as Voronoi regions, Minkowski sums, and linear and quadratic programming.Of utmost importance to programmers but rarely discussed in this much detail in other books are the chapters covering numerical and geometric robustness, both essential topics for collision detection systems. Also unique are the chapters discussing how graphics hardware can assist in collision detection computations and on advanced optimization for modern computer architectures. All in all, this comprehensive book will become the industry standard for years to come.