Designing trajectories for manipulation through contact is challenging as it requires reasoning of object & robot trajectories as well as complex contact sequences simultaneously. In this paper, we present a novel framework for simultaneously designing trajectories of robots, objects, and contacts efficiently for contact-rich manipulation. We propose a hierarchical optimization framework where Mixed-Integer Linear Program (MILP) selects optimal contacts between robot & object using approximate dynamical constraints, and then a NonLinear Program (NLP) optimizes trajectory of the robot(s) and object considering full nonlinear constraints. We present a convex relaxation of bilinear constraints using binary encoding technique such that MILP can provide tighter solutions with better computational complexity. The proposed framework is evaluated on various manipulation tasks where it can reason about complex multi-contact interactions while providing computational advantages. We also demonstrate our framework in hardware experiments using a bimanual robot system.

Linear programming (LP) relaxations are widely employed in exact solution methods for multilinear programs (MLPs). These relaxations can be obtained by using recursive McCormick linearizations (RMLs), by which an MLP is linearized by iteratively substituting bilinear products with artificial variables and constraints. This article introduces a systematic approach to identifying RMLs. We focus on identifying RMLs with a small number of artificial variables and strong LP bounds. We present a novel mechanism for representing all the possible RMLs, which we use to design an exact mixed-integer programming (MIP) formulation to identify minimum-size RMLs; this problem is NP-hard in general, but we show that it is fixed-parameter tractable if each monomial is composed of at most three variables. Moreover, we explore the structural properties of our formulation to derive an exact MIP model that identifies RMLs of a given size with the best-possible LP relaxation bound. We test our algorithms by conducting numerical experiments on a large collection of MLPs. Numerical results indicate that the RMLs obtained with our algorithms can be significantly smaller than those derived from heuristic or greedy approaches, leading, in many cases, to tighter LP relaxation bounds. Moreover, our linearization strategies can be used to reformulate MLPs as quadratically constrained programs (QCPs), which can then be efficiently solved using state-of-the-art solvers for QCPs. This QCP-based solution approach is highly beneficial for hard MLP instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete.

Abstract When computing bounds, spatial branch-and-bound algorithms often linearly outer approximate convex relaxations for non-convex expressions in order to capitalize on the efficiency and robustness of linear programming solvers. Considering that linear outer approximations sacrifice accuracy when approximating highly nonlinear functions and recognizing the recent advancements in the efficiency and robustness of available methods to solve optimization problems with quadratic objectives and constraints, we contemplate here the construction of quadratic outer approximations of twice-differentiable convex functions for use in deterministic global optimization. To this end, we present a novel cutting-plane algorithm that determines the tightest scaling parameter, $$\alpha $$ α , in the second-order Taylor series approximation quadratic underestimator proposed by Su et al. [25]. We use a representative set of convex functions extracted from optimization benchmark libraries to showcase–qualitatively and quantitatively–the tightness of the constructed quadratic underestimators and to demonstrate the overall computational efficiency of our algorithm. Furthermore, we extend our construction procedure to generate even tighter quadratic underestimators by allowing overestimation in infeasible polyhedral regions of optimization problems, as informed by the latter’s linear constraints.

Abstract Recent advances in the efficiency and robustness of algorithms solving convex quadratically constrained quadratic programming (QCQP) problems motivate developing techniques for creating convex quadratic relaxations that, although more expensive to compute, provide tighter bounds than their classical linear counterparts. In the first part of this two-paper series (Strahl et al. Constructing tight quadratic relaxations for global optimization: I. Outer-approximating twice-differentiable convex functions. Forthcoming, (2024)), we developed a cutting-plane algorithm to construct convex quadratic underestimators for twice-differentiable convex functions, which we extend here to address the case of non-convex difference-of-convex (d.c.) functions as well. Furthermore, we generalize our approach to consider a hierarchy of quadratic forms, thereby allowing the construction of even tighter underestimators. Utilizing a benchmark library of d.c. functions, we demonstrate noteworthy reduction in the hypervolume between our quadratic underestimators and linear ones constructed at the same points. Additionally, we construct convex QCQP relaxations at the root node of a spatial branch-and-bound tree for a set of systematically created d.c. optimization problems in up to four dimensions, and we show that our relaxations reduce the gap between the lower bound computed by the state-of-the-art global optimization solver BARON and the optimal solution by an excess of 90%, on average.

Contacts are central to most manipulation tasks as they provide additional dexterity to robots to perform challenging tasks. However, frictional contacts leads to complex complementarity constraints. Planning in the presence of contacts requires robust handling of these constraints to find feasible solutions. This paper presents PyRoboCOP which is a lightweight Python-based package for control and optimization of robotic systems described by nonlinear Differential Algebraic Equations (DAEs). In particular, the proposed optimization package can handle systems with contacts that are described by complementarity constraints. We also present a general framework for specifying obstacle avoidance constraints using complementarity constraints. The package performs direct transcription of the DAEs into a set of nonlinear equations by performing orthogonal collocation on finite elements. The resulting optimization problem belongs to the class of Mathematical Programs with Complementarity Constraints (MPCCs). MPCCs fail to satisfy commonly assumed constraint qualifications and require special handling of the complementarity constraints in order for NonLinear Program (NLP) solvers to solve them effectively. PyRoboCOP provides automatic reformulation of the complementarity constraints that enables NLP solvers to perform optimization of robotic systems. The package is interfaced with for obtaining sparse derivatives by automatic differentiation and for performing optimization. We provide extensive numerical examples for various different robotic systems with collision avoidance as well as contact constraints represented using complementarity constraints. We provide comparisons with other open source optimization packages like and . The code is open sourced and available at https://github.com/merlresearch/PyRoboCOP.Note to Practitioners—PyRoboCOP is intended to be an easy-to-use software package written in Python which can be used for optimization, estimation and control for a large class of robotic systems. Including, in particular, contact-rich applications to deal with complex scenarios that arise when making and breaking contacts during a task. Typical problems that can be solved with our work are trajectory and control sequence optimization, parameter estimation. To make the proposed software package easier for practitioners, the paper provides access to the package and a large number of example problems. Furthermore, the package also provides a guide describing the details of all the methods a user might have to implement for their own system. Compared to some of the other packages, PyRoboCOP works with NumPy object arrays which is the native computing package in Python. We believe that this will make it much easier to learn and use compared to some of the other optimal control packages.

Contact-implicit trajectory optimization (CITO) is an effective method to plan complex trajectories for various contact-rich systems including manipulation and locomotion. CITO formulates a mathematical program with complementarity constraints (MPCC) that enforces that contact forces must be zero when points are not in contact. However, MPCC solve times increase steeply with the number of allowable points of contact, which limits CITO's applicability to problems in which only a few, simple geometries are allowed us to make contact. This article introduces simultaneous trajectory optimization and contact selection (STOCS), as an extension of CITO that overcomes this limitation. The innovation of STOCS is to identify salient contact points and times inside the iterative trajectory optimization process. This effectively reduces the number of variables and constraints in each MPCC invocation. The STOCS framework, instantiated with key contact identification subroutines, renders the optimization of manipulation trajectories computationally tractable even for high-fidelity geometries consisting of tens of thousands of vertices.

Ride-hailing services have expanded the role of shared mobility in passenger transportation systems, creating new markets and creative planning solutions for major urban centers. In this paper, we consider their use for the first-mile or last-mile passenger transportation in coordination with a mass transit service to provide a seamless multimodal transportation experience for the user. A system that provides passengers with predictable information on travel and waiting times in their commutes is immensely valuable. We envision that the passengers will inform the system of their desired travel and arrival windows so that the system can jointly optimize the schedules of passengers. The problem we study balances minimizing travel time and the number of trips taken by the last-mile vehicles, so that long-term planning, maintenance, and environmental impact are all taken into account. We focus on the case where the last-mile service aggregates passengers by destination. We show that this problem is NP-hard, and we propose a decision diagram–based branch-and-price decomposition model that can solve instances of real-world size (10,000 passengers spread over an hour, 50 last-mile destinations, 600 last-mile vehicles) in computational time (∼1 minute) that is orders of magnitude faster than the solution times of other methods appearing in the literature. Our experiments also indicate that aggregating passengers by destination on the last-mile service provides high-quality solutions to more general settings. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods and Analysis. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2019.0163 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2019.0163 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Generalizable manipulation requires that robots be able to interact with novel objects and environment. This requirement makes manipulation extremely challenging as a robot has to reason about complex frictional interactions with uncertainty in physical properties of the object and the environment. In this article, we study robust optimization for planning of pivoting manipulation in the presence of uncertainties. We present insights about how friction can be exploited to compensate for inaccuracies in the estimates of the physical properties during manipulation. Under certain assumptions, we derive analytical expressions for stability margin provided by friction during pivoting manipulation. This margin is then used in a contact implicit bilevel optimization framework to optimize a trajectory that maximizes this stability margin to provide robustness against uncertainty in several physical parameters of the object. We present analysis of the stability margin with respect to several parameters involved in the underlying bilevel optimization problem. We demonstrate our proposed method using a 6 DoF manipulator for manipulating several different objects. We also design and validate an MPC controller using the proposed algorithm which can track and regulate the position of the object during manipulation.

Planning and control for uncertain contact systems is challenging as it is not clear how to propagate uncertainty for planning. Contact-rich tasks can be modeled efficiently using complementarity constraints among other techniques. In this paper, we present a stochastic optimization technique with chance constraints for systems with stochastic complementarity constraints. We use a particle filter-based approach to propagate moments for stochastic complementarity system. To circumvent the issues of open-loop chance constrained planning, we propose a contact-aware controller for covariance steering of the complementarity system. Our optimization problem is formulated as Non-Linear Programming (NLP) using bilevel optimization. We present an important-particle algorithm for numerical efficiency for the underlying control problem. We verify that our contact-aware closed-loop controller is able to steer the covariance of the states under stochastic contact-rich tasks.

Humans can effortlessly perform very complex, dexterous manipulation tasks by reacting to sensor observations. In contrast, robots can not perform reactive manipulation and they mostly operate in open-loop while interacting with their environment. Consequently, the current manipulation algorithms either are inefficient in performance or can only work in highly structured environments. In this paper, we present closed-loop control of a complex manipulation task where a robot uses a tool to interact with objects. Manipulation using a tool leads to complex kinematics and contact constraints that need to be satisfied for generating feasible manipulation trajectories. We first present an open-loop controller design using Non-Linear Programming (NLP) that satisfies these constraints. In order to design a closed-loop controller, we present a pose estimator of objects and tools using tactile sensors. Using our tactile estimator, we design a closed-loop controller based on Model Predictive Control (MPC). The proposed algorithm is verified using a 6 DoF manipulator on tasks using a variety of objects and tools. We verify that our closed-loop controller can successfully perform tool manipulation under several unexpected contacts.

Dynamic movement primitives are widely used for learning skills that can be demonstrated to a robot by a skilled human or controller. While their generalization capabilities and simple formulation make them very appealing to use, they possess no strong guarantees to satisfy operational safety constraints for a task. We present constrained dynamic movement primitives (CDMPs), which can allow for positional constraint satisfaction in the robot workspace. Our method solves a non-linear optimization to perturb an existing DMP's forcing weights to admit a Zeroing Barrier Function (ZBF), which certifies positional workspace constraint satisfaction. We demonstrate our approach under different positional constraints on the end-effector movement on multiple physical robots, such as obstacle avoidance and workspace limitations.

PyROBOCOP is a Python-based package for control, optimization and estimation of robotic systems described by nonlinear Differential Algebraic Equations (DAEs). In particular, the package can handle systems with contacts that are described by complementarity constraints and provides a general framework for specifying obstacle avoidance constraints. The package performs direct transcription of the DAEs into a set of nonlinear equations by performing orthogonal collocation on finite elements. PyROBOCOP provides automatic reformulation of the complementarity constraints that are tractable to NLP solvers to perform optimization of robotic systems. The package is interfaced with ADOL-C [1] for obtaining sparse derivatives by automatic differentiation and IPOPT [2] for performing optimization. We evaluate PyROBOCOP on several manipulation problems for control and estimation.

Generalizable manipulation requires that robots be able to interact with novel objects and environment. This requirement makes manipulation extremely challenging as a robot has to reason about complex frictional interaction with uncertainty in physical properties of the object. In this paper, we study robust optimization for control of pivoting manipulation in the presence of uncertainties. We present insights about how friction can be exploited to compensate for the inaccuracies in the estimates of the physical properties during manipulation. In particular, we derive analytical expressions for stability margin provided by friction during pivoting manipulation. This margin is then used in a bilevel trajectory optimization algorithm to design a controller that maximizes this stability margin to provide robustness against uncertainty in physical properties of the object. We demonstrate our proposed method using a 6 DoF manipulator for manipulating several different objects.

Business research practice is witnessing a surge in the integration of predictive modeling and prescriptive analysis. We describe a modeling framework JANOS that seamlessly integrates the two streams of analytics, allowing researchers and practitioners to embed machine learning models in an end-to-end optimization framework. JANOS allows for specifying a prescriptive model using standard optimization modeling elements such as constraints and variables. The key novelty lies in providing modeling constructs that enable the specification of commonly used predictive models within an optimization model, have the features of the predictive model as variables in the optimization model, and incorporate the output of the predictive models as part of the objective. The framework considers two sets of decision variables: regular and predicted. The relationship between the regular and the predicted variables is specified by the user as pretrained predictive models. JANOS currently supports linear regression, logistic regression, and neural network with rectified linear activation functions. In this paper, we demonstrate the flexibility of the framework through an example on scholarship allocation in a student enrollment problem and provide a numeric performance evaluation. Summary of Contribution. This paper describes a new software tool, JANOS, that integrates predictive modeling and discrete optimization to assist decision making. Specifically, the proposed solver takes as input user-specified pretrained predictive models and formulates optimization models directly over those predictive models by embedding them within an optimization model through linear transformations.

Quantum annealing (QA) can be used to quickly obtain near-optimal solutions for quadratic unconstrained binary optimization (QUBO) problems. In QA hardware, each decision variable of a QUBO should be mapped to one or more adjacent qubits in such a way that pairs of variables defining a quadratic term in the objective function are mapped to some pair of adjacent qubits. However, qubits have limited connectivity in existing QA hardware. This has spurred work on preprocessing algorithms for embedding the graph representing problem variables with quadratic terms into the hardware graph representing qubits adjacencies, such as the Chimera graph in hardware produced by D-Wave Systems. In this paper, we use integer linear programming to search for an embedding of the problem graph into certain classes of minors of the Chimera graph, which we call template embeddings. One of these classes corresponds to complete bipartite graphs, for which we show the limitation of the existing approach based on minimum odd cycle transversals (OCTs). One of the formulations presented is exact and thus can be used to certify the absence of a minor embedding using that template. On an extensive test set consisting of random graphs from five different classes of varying size and sparsity, we can embed more graphs than a state-of-the-art OCT-based approach, our approach scales better with the hardware size, and the runtime is generally orders of magnitude smaller. Summary of Contribution: Our work combines classical and quantum computing for operations research by showing that integer linear programming can be successfully used as a preprocessing step for adiabatic quantum optimization. We use it to determine how a quadratic unconstrained binary optimization problem can be solved by a quantum annealer in which the qubits are coupled as in a Chimera graph, such as in the quantum annealers currently produced by D-Wave Systems. The paper also provides a timely introduction to adiabatic quantum computing and related work on minor embeddings.

This article presents a data-driven method for computing reachable sets where active learning (AL) is used to reduce the computational burden. Set-based methods used to estimate reachable sets typically do not scale well with the state-space dimension, or rely heavily on the existence of a model. If such a model is not available, it is simple to generate state trajectory data by numerically simulating black-box oracles of systems (whose dynamics are unknown) from sampled initial conditions. Using these data samples, the estimation of reachable sets can be posed as a classification problem, wherein AL can intelligently select samples that are most informative and least similar to previously labeled samples. By exploiting submodularity, the actively learned samples can be selected efficiently, with bounded suboptimality. Our proposed framework is illustrated by estimating the domains of attractions of model predictive controllers (MPCs) and reinforcement learners. We also consider a scenario where there are two oracles that differ with respect to evaluation costs and labeling accuracy. We propose a framework to reduce the dependency of the expensive oracle in labeling samples using disagreement-based AL (DBAL). The potential of the DBAL algorithm is demonstrated on a solver selection problem for real-time MPC.

The goal of this paper is to present a method for simultaneous trajectory and local stabilizing policy optimization to generate local policies for trajectory-centric model-based reinforcement learning (MBRL). This is motivated by the fact that global policy optimization for non-linear systems could be a very challenging problem both algorithmically and numerically. However, a lot of robotic manipulation tasks are trajectory-centric, and thus do not require a global model or policy. Due to inaccuracies in the learned model estimates, an open-loop trajectory optimization process mostly results in very poor performance when used on the real system. Motivated by these problems, we try to formulate the problem of trajectory optimization and local policy synthesis as a single optimization problem. It is then solved simultaneously as an instance of nonlinear programming. We provide some results for analysis as well as achieved performance of the proposed technique under some simplifying assumptions.

Computing Nash equilibrium (NE) of multi-player games has witnessed renewed interest due to recent advances in generative adversarial networks. However, computing equilibrium efficiently is challenging. To this end, we introduce the Gradient-based Nikaido-Isoda (GNI) function which serves: (i) as a merit function, vanishing only at the first-order stationary points of each player’s optimization problem, and (ii) provides error bounds to a stationary Nash point. Gradient descent is shown to converge sublinearly to a first-order stationary point of the GNI function. For the particular case of bilinear min-max games and multi-player quadratic games, the GNI function is convex. Hence, the application of gradient descent in this case yields linear convergence to an NE (when one exists). In our numerical experiments, we observe that the GNI formulation always converges to the first-order stationary point of each player’s optimization problem.