Kodiak: An Implementation Framework for Branch and Bound Algorithms

Recursive branch and bound algorithms are often used to refine and isolate solutions to several classes of global optimization problems. A rigorous computation framework for the solution of systems of equations and inequalities involving nonlinear real arithmetic over hyper-rectangular variable and parameter domains is presented. It is derived from a generic branch and bound algorithm that has been formally verified, and utilizes self-validating enclosure methods, namely interval arithmetic and, for polynomials and rational functions, Bernstein expansion. Since bounds computed by these enclosure methods are sound, this approach may be used reliably in software verification tools. Advantage is taken of the partial derivatives of the constraint functions involved in the system, firstly to reduce the branching factor by the use of bisection heuristics and secondly to permit the computation of bifurcation sets for systems of ordinary differential equations. The associated software development, Kodiak, is presented, along with examples of three different branch and bound problem types it implements

Experiments with hybrid Bernstein global optimization algorithm for the OPF problem in power systems

This paper presents an algorithm based on the Bernstein form of polynomials for solving the optimal power flow (OPF) problem in electrical power networks. The proposed algorithm combines local and global optimization methods and is therefore referred to as a `hybrid' Bernstein algorithm in the context of this work. The proposed algorithm is a branch-and-bound (B&B) procedure wherein a local search method is used to obtain a good upper bound on the global minimum at each branching node. Subsequently, the Bernstein form of polynomials is used to obtain a lower bound on the global minimum. The performance of the proposed algorithm is compared with the previously reported Bernstein algorithm to demonstrate its effi cacy in terms of the chosen performance metrics. Furthermore, the proposed algorithm is tested by solving the OPF problem for several benchmark IEEE power system examples and its performance is compared with generic global optimization solvers such as BARON and COUENNE. The test results demonstrate that the algorithm HBBB delivers satisfactory performance in terms of solution optimality.This research is supported by the National Research Foundation, Prime Ministers Office, Singapore, under its CREATE programme

Position analysis based on multi-affine formulations

Aplicat embargament des de la data de defensa fins el 31/5/2022The position analysis problem is a fundamental issue that underlies many problems in Robotics such as the inverse kinematics of serial robots, the forward kinematics of parallel robots, the coordinated manipulation of objects, the generation of valid grasps, the constraint-based object positioning, the simultaneous localization and map building, and the analysis of complex deployable structures. It also arises in other fields, such as in computer aided design, when the location of objects in a design is given in terms of geometric constrains, or in the conformational analysis of biomolecules. The ubiquity of this problem, has motivated an intense quest for methods able of tackling it. Up to now, efficient algorithms for the general problem have remained elusive and they are only available for particular cases. Moreover, the complexity of the problem has typically led to methods difficult to be implemented. Position analysis can be decomposed into two equally important steps: obtaining a set of closure equations, and solving them. This thesis deals with both of them to obtain a general, simple, and yet efficient solution method that we call the trapezoid method. The first step is addressed relying on dual quaternions. Although it has not been properly highlighted in the past, the use of dual quaternions permits expressing the closure condition of a kinematic loop involving only lower pairs as a system of multi-affine equations. In this thesis, this property is leveraged to introduce an interval-based method specially tailored for solving multi-affine systems. The proposed method is objectively simpler (in the sense that it is easier to understand and to implement) than previous methods based on general techniques such as interval Newton methods, conversions to Bernstein basis, or linear relaxations. Moreover, it relies on two simple operations, namely, linear interpolations and projections on coordinate planes, which can be executed with a high performance. The result is a method that accurately and efficiently bounds the valid solutions of the problem at hand. To further improve the accuracy, we propose the use of redundant, multi affine equations that are derived from the minimal set of equations describing the problem. To improve the efficiency, we introduce a variable elimination methodology that preserves the multi-affinity of the system of equations. The generality and the performance of the proposed trapezoid method are extensively evaluated on different kind of mechanisms, including spherical mechanisms, generic 6R and 7R loops, over-constrained systems, and multi-loop mechanisms. The proposed method is, in all cases, significantly faster than state of the art alternatives.El problema de l'anàlisi de posició és un tema fonamental que subjau a molts problemes de la robòtica, com ara la cinemàtica inversa de robots sèrie, la cinemàtica directa de robots paral·lels, la manipulació coordinada d'objectes, la generació de prensions vàlides amb mans robòtiques, el posicionament d'objectes basat en restriccions, la localització i la creació de mapes de forma simultània, i l'anàlisi d'estructures desplegables complexes. També sorgeix en altres camps, com ara en el disseny assistit per ordinador, quan la ubicació dels objectes en un disseny es dóna en termes de restriccions geomètriques o en l'anàlisi conformacional de biomolècules. La omnipresència d'aquest problema ha motivat una intensa recerca de mètodes capaços d'afrontar-lo. Fins al moment, els algoritmes eficients per al problema general han estat esquius i només estan disponibles per a casos particulars. A més, la complexitat del problema normalment ha conduït a mètodes difícils d'implementar. L'anàlisi de posició es pot descompondre en dos passos igualment importants: l'obtenció d'un sistema d'equacions de tancament i la resolució d'aquest sistema. Aquesta tesi tracta de tots dos passos per tal d'obtenir un mètode de solució general, senzill i alhora eficient que anomenem el mètode del trapezoide. El primer pas s'aborda utilitzant quaternions duals. Tot i que no ha estat suficientment destacat en el passat, l'ús de quaternions duals permet expressar la condició de tancament d'un bucle cinemàtic que impliqui només parells inferiors com a un sistema d'equacions multi-afins. En aquesta tesi s'aprofita aquesta propietat per introduir un mètode especialment dissenyat per resoldre sistemes multi-afins. El mètode proposat és objectivament més senzill (en el sentit que és més fàcil d'entendre i d'implementar) que els mètodes anteriors que utilitzen tècniques generals com ara els mètodes de Newton basats en intervals, les conversions a la base de Bernstein o les relaxacions lineals. A més, el mètode es basa en dues operacions simples, a saber, les interpolacions lineals i les projeccions en plans de coordenades, que es poden executar de forma molt eficient. El resultat és un mètode que acota amb precisió i eficiència les solucions vàlides del problema. Per millorar encara més la precisió, proposem l'ús d'equacions multi-afins redundants derivades del conjunt mínim d'equacions que descriuen el problema. Per altra banda, per millorar l'eficiència, introduïm un metodologia d'eliminació de variables que preserva la multi-afinitat del sistema d'equacions. La generalitat i el rendiment del mètode del trapezoide s'avalua extensivament en diferents tipus de mecanismes, inclosos els mecanismes esfèrics, bucles 6R i 7R genèrics, sistemes sobre-restringits i mecanismes de múltiples bucles. El mètode proposat és, en tots els casos, significativament més ràpid que els mètodes alternatius descrits en la literatura fins al moment.Postprint (published version

Fast, Optimal, and Safe Motion Planning for Bipedal Robots

Bipedal robots have the potential to traverse a wide range of unstructured environments, which are otherwise inaccessible to wheeled vehicles. Though roboticists have successfully constructed controllers for bipedal robots to walk over uneven terrain such as snow, sand, or even stairs, it has remained challenging to synthesize such controllers in an online fashion while guaranteeing their satisfactory performance. This is primarily due to the lack of numerical method that can accommodate the non-smooth dynamics, high degrees of freedom, and underactuation that characterize bipedal robots. This dissertation proposes and implements a family of numerical methods that begin to address these three challenges along three dimensions: optimality, safety, and computational speed. First, this dissertation develops a convex relaxation-based approach to solve optimal control for hybrid systems without a priori knowledge of the optimal sequence of transition. This is accomplished by formulating the problem in the space of relaxed controls, which gives rise to a linear program whose solution is proven to compute the globally optimal controller. This conceptual program is solved using a sequence of semidefinite programs whose solutions are proven to converge from below to the true optimal solution of the original optimal control problem. Moreover, a method to synthesize the optimal controller is developed. Using an array of examples, the performance of this method is validated on problems with known solutions and also compared to a commercial solver. Second, this dissertation constructs a method to generate safety-preserving controllers for a planar bipedal robot walking on flat ground by performing reachability analysis on simplified models under the assumption that the difference between the two models can be bounded. Subsequently, this dissertation describes how this reachable set can be incorporated into a Model Predictive Control framework to select controllers that result in safe walking on the biped in an online fashion. This method is validated on a 5-link planar model. Third, this dissertation proposes a novel parallel algorithm capable of finding guaranteed optimal solutions to polynomial optimization problems up to pre-specified tolerances. Formal proofs of bounds on the time and memory usage of such method are also given. Such algorithm is implemented in parallel on GPUs and compared against state-of-the-art solvers on a group of benchmark examples. An application of such method on a real-time trajectory-planning task of a mobile robot is also demonstrated. Fourth, this dissertation constructs an online Model Predictive Control framework that guarantees safety of a 3D bipedal robot walking in a forest of randomly-placed obstacles. Using numerical integration and interval arithmetic techniques, approximations to trajectories of the robot are constructed along with guaranteed bounds on the approximation error. Safety constraints are derived using these error bounds and incorporated in a Model Predictive Control framework whose feasible solutions keep the robot from falling over and from running into obstacles. To ensure that the bipedal robot is able to avoid falling for all time, a finite-time terminal constraint is added to the Model Predictive Control algorithm. The performance of this method is implemented and compared against a naive Model Predictive Control method on a biped model with 20 degrees of freedom. In summary, this dissertation presents four methods for control synthesis of bipedal robots with improvements in either optimality, safety guarantee, or computational speed. Furthermore, the performance of all proposed methods are compared with existing methods in the field.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/162880/1/pczhao_1.pd

Fast and Safe Trajectory Optimization for Autonomous Mobile Robots using Reachability Analysis

Autonomous mobile robots (AMRs) can transform a wide variety of industries including transportation, shipping and goods delivery, and defense. AMRs must match or exceed human performance in metrics for task completion and safety. Motion plans for AMRs are generated by solving an optimization program where collision avoidance and the trajectory obeying a dynamic model of the robot are enforced as constraints. This dissertation focuses on three main challenges associated with trajectory planning. First, collision checks are typically performed at discrete time steps. Second, there can be a nontrivial gap between the planning model used and the actual system. Finally, there is inherent uncertainty in the motion of other agents or robots. This dissertation first proposes a receding-horizon planning methodology called Reachability-based Trajectory Design (RTD) to address the first and second challenges, where uncertainty is dealt with robustly. Sums-of-Squares (SOS) programming is used to represent the forward reachable set for a dynamic system plus uncertainty, over an interval of time, as a polynomial level set. The trajectory optimization is a polynomial optimization program over a space of trajectory parameters. Hardware demonstrations are implemented on a Segway, rover, and electric vehicle. In a simulation of 1,000 trials with static obstacles, RTD is compared to Rapidly-exploring Random Tree (RRT) and Nonlinear Model Predictive Control (NMPC) planners. RTD has success rates of 95.4% and 96.3% for the Segway and rover respectively, compared to 97.6% and 78.2% for RRT and 0% for NMPC planners. RTD is the only successful planner with no collisions. In 10 simulations with a CarSim model, RTD navigates a test track on all trials. In 1,000 simulations with random dynamic obstacles RTD has success rates of 96.8% and 100% respectively for the electric vehicle and Segway, compared to 77.3% and 92.4% for a State Lattice planner. In 100 simulations performing left turns, RTD has a success rate of 99% compared to 80% for an MPC controller tracking the lane centerline. The latter half of the dissertation treats uncertainty with the second and/or third challenges probabilistically. The Chance-constrained Parallel Bernstein Algorithm (CCPBA) allows one to solve the trajectory optimization program from RTD when obstacle states are given as probability functions. A comparison for an autonomous vehicle planning a lane change with one obstacle shows an MPC algorithm using Cantelli's inequality is unable to find a solution when the obstacle's predictions are generated with process noise three orders of magnitude less than CCPBA. In environments with 1-6 obstacles, CCPBA finds solutions in 1e-3 to 1.2 s compared to 1 to 16 s for an NMPC algorithm using the Chernoff bound. A hardware demonstration is implemented on the Segway. The final portion of the dissertation presents a chance-constrained NMPC method where uncertain components of the robot model are estimated online. The application is an autonomous vehicle with varying road surfaces. In the first study, the controller uses a linear tire force model. Over 200 trials of lane changes at 17 m/s, the chance-constrained controller has a cost 86% less than a controller using fixed coefficients for snow, and only 29% more than an oracle controller using the simulation model. The chance-constrained controller also has 0 lateral position constraint violations, while an adaptive-only controller has minor violations. The second study uses nonlinear tire models on a more aggressive maneuver and provides similar results.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169729/1/skvaskov_1.pd

A New Approach to CNC Programming of Plunge Milling

ABSTRACT A New Approach to CNC Programming of Plunge Milling Sherif Abdelkhalek, PhD. Concordia University, 2013. In current industrial applications many engineering parts are made of hard materials including dies, mold cavities and aerospace parts. Manufacturing these types of parts is classified as pocket milling. By using the regular machining methods, pocket milling takes a long time accompanied by high cost. Plunge milling, is a new machining strategy that has proven to have an excellent performance in the rough machining of hard materials. In plunge milling, the cutter is fed in the direction of the spindle axis, with the highest structural rigidity which showed a very interesting performance in removing the excess material rapidly in the rough operations. Mainly, according to the previous researchers, two directions are adopted to improve the efficiency of the plunge milling process. First, to reduce the cutting forces and increase chatter stability which attracts the majority of the researchers. Second, to optimize the tool path planning which has less attention. Therefore, in the first part of the research, a new practical approach is established in optimized procedures to generate the tool paths for plunge milling of pockets, even for these with free-form boundaries and islands. This innovative approach is proposed as follows: (1) fill a pocket with minimum number of specified radii circles which are tangent to each other and/or the pocket boundary without overlapping by building an algorithm using the maximum hole degree (MHD) theory for solving the circle packing problem. (2) cover the areas left between the non-overlapped circles by the same used specified radii. Finally, solve the travelling sales man problem (TSP) for the circles with the same radii by using the simulated annealing algorithm. According to the results, this approach significantly advances the tool path planning technique for pockets plunge milling. In the second part of the research, a new algorithm is proposed to calculate the global solution for constraint polynomial functions by using subtractive clustering which makes the results more accurate and faster to be obtained. This part is extremely useful to calculate the depth of cut for each plunging place in case of having a polynomial surface as a bottom of the machined pocket with high accuracy, and less calculation time to avoid gauging between the tool and the bottom surface. The polynomial function can be classified according to the number of variables. In the proposed research, the functions with one and two variables have more importance because they graphically represent curves and surfaces which are the cases under study. Since the polynomial function under study can be represented graphically according to the number of the variables, the change in the function’s shape can be detected by the feature recognition. The feature recognition is done for the function’s shape by calculating the surface or curve curvature at the data points. The main procedure is; (1) identifying the entire features of the objective function which are classified according to the curvature as convex, concave, plane, and hyperbolic, (2) applying the sub-clustering technique for convex and concave regions to find the approximated centers of these regions, and eventually, (3) the clusters’ centers are calculated and used as initial points for local optimization technique which gives the local critical point for each region. The local minima are calculated, the global minimum is the minimum of the local minima

New Techniques for the Modeling, Processing and Visualization of Surfaces and Volumes

With the advent of powerful 3D acquisition technology, there is a growing demand for the modeling, processing, and visualization of surfaces and volumes. The proposed methods must be efficient and robust, and they must be able to extract the essential structure of the data and to easily and quickly convey the most significant information to a human observer. Independent of the specific nature of the data, the following fundamental problems can be identified: shape reconstruction from discrete samples, data analysis, and data compression. This thesis presents several novel solutions to these problems for surfaces (Part I) and volumes (Part II). For surfaces, we adopt the well-known triangle mesh representation and develop new algorithms for discrete curvature estimation,detection of feature lines, and line-art rendering (Chapter 3), for connectivity encoding (Chapter 4), and for topology preserving compression of 2D vector fields (Chapter 5). For volumes, that are often given as discrete samples, we base our approach for reconstruction and visualization on the use of new trivariate spline spaces on a certain tetrahedral partition. We study the properties of the new spline spaces (Chapter 7) and present efficient algorithms for reconstruction and visualization by iso-surface rendering for both, regularly (Chapter 8) and irregularly (Chapter 9) distributed data samples

Rigorous solution techniques for numerical constraint satisfaction problems

A constraint satisfaction problem (e.g., a system of equations and inequalities) consists of a finite set of constraints specifying which value combinations from given variable domains are admitted. It is called numerical if its variable domains are continuous. Such problems arise in many applications, but form a difficult problem class since they are NP-hard. Solving a constraint satisfaction problem is to find one or more value combinations satisfying all its constraints. Numerical computations on floating-point numbers in computers often suffer from rounding errors. The rigorous control of rounding errors during numerical computations is highly desired in many applications because it would benefit the quality and reliability of the decisions based on the solutions found by the computations. Various aspects of rigorous numerical computations in solving constraint satisfaction problems are addressed in this thesis: search, constraint propagation, combination of inclusion techniques, and post-processing. The solution of a constraint satisfaction problem is essentially performed by a search. In this thesis, we propose a new complete search technique (i.e., it can find all solutions within a predetermined tolerance) for numerical constraint satisfaction problems. This technique is general and can be used in place of branching steps in most branch-and-prune methods. Moreover, this new technique speeds up the most recent general search strategy (often by an order of magnitude) and provides a concise representation of solutions. To make a constraint satisfaction problem easier to solve, a major approach, called constraint propagation, in the constraint programming1 field is often used to reduce the variable domains (by discarding redundant value combinations from the domains). Basing on directed acyclic graphs, we propose a new constraint propagation technique and a method for coordinating constraint propagation and search. More importantly, we propose a novel generic scheme for combining multiple inclusion techniques2 in numerical constraint propagation. This scheme allows bringing into the constraint propagation framework the strengths of various techniques coming from different fields. To illustrate the flexibility and efficiency of the generic scheme, we base on this scheme and devise several specific combination strategies for rigorous numerical constraint propagation using interval constraint propagation, interval arithmetic, affine arithmetic, and linear programming. Our experiments show that the new propagation techniques outperform previously available methods by 1 to 4 orders of magnitude or more in speed. We also propose several post-processing techniques for the representation of continuums of solutions. Based on connectedness, they allow grouping each cluster of connected solution subsets into a larger subset, thus allowing getting additional grouping information. Potentially, these techniques enable interval-based solution techniques to be alternatives to bounding-volume techniques in applications such as collision detection and interactive graphics. __________________________________________________ 1 Constraint programming is an approach to programming that relies on both reasoning and computing. 2 An inclusion technique is to include a set of interest into enclosures. It is also called an enclosure technique