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Models, Feedback Control, and Open Problems of 3D Bipedal Robot Walking
, 2014
"... The fields of control and robotics are working toward the development of bipedal robots that can realize walking motions with the stability and agility of a human being. Dynamic models for bipeds are hybrid in nature. They contain both continuous and discrete elements, with switching events that are ..."
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Cited by 5 (3 self)
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The fields of control and robotics are working toward the development of bipedal robots that can realize walking motions with the stability and agility of a human being. Dynamic models for bipeds are hybrid in nature. They contain both continuous and discrete elements, with switching events that are governed by a combination of unilateral constraints and impulselike forces that occur at foot touchdown. Control laws for these machines must be hybrid as well. The goals of this paper are fourfold: highlight certain properties of the models which greatly influence the control law design; overview the literature; present two control design approaches in depth; and indicate some of the many open problems.
Convex Optimization of Nonlinear Feedback Controllers via Occupation Measures
"... Abstract—In this paper, we present an approach for designing feedback controllers for polynomial systems that maximize the size of the timelimited backwards reachable set (BRS). We rely on the notion of occupation measures to pose the synthesis problem as an infinite dimensional linear program (LP) ..."
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Cited by 4 (1 self)
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Abstract—In this paper, we present an approach for designing feedback controllers for polynomial systems that maximize the size of the timelimited backwards reachable set (BRS). We rely on the notion of occupation measures to pose the synthesis problem as an infinite dimensional linear program (LP) and provide finite dimensional approximations of this LP in terms of semidefinite programs (SDPs). The solution to each SDP yields a polynomial control policy and an outer approximation of the largest achievable BRS. In contrast to traditional Lyapunov based approaches which are nonconvex and require feasible initialization, our approach is convex and does not require any form of initialization. The resulting timevarying controllers and approximated reachable sets are wellsuited for use in a trajectory library or feedback motion planning algorithm. We demonstrate the efficacy and scalability of our approach on four nonlinear systems. I.
Lyapunov Analysis of Rigid Body Systems with Impacts and Friction via SumsofSquares
"... Many critical tasks in robotics, such as locomotion or manipulation, involve collisions between a rigid body and the environment or between multiple bodies. Sumsofsquares (SOS) based methods for numerical computation of Lyapunov certificates are a powerful tool for analyzing the stability of conti ..."
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Many critical tasks in robotics, such as locomotion or manipulation, involve collisions between a rigid body and the environment or between multiple bodies. Sumsofsquares (SOS) based methods for numerical computation of Lyapunov certificates are a powerful tool for analyzing the stability of continuous nonlinear systems, which can play a powerful role in motion planning and control design. Here, we present a method for applying sumsofsquares verification to rigid bodies with Coulomb friction undergoing discontinuous, inelastic impact events. The proposed algorithm explicitly generates Lyapunov certificates for stability, positive invariance, and reachability over admissible (nonpenetrating) states and contact forces. We leverage the complementarity formulation of contact, which naturally generates the semialgebraic constraints that define this admissible region. The approach is demonstrated on multiple robotics examples, including simple models of a walking robot and a perching aircraft.
Control and verification of highdimensional systems via DSOS and SDSOS optimization
 In Proceedings of the 53rd IEEE Conference on Decision and Control
, 2014
"... Abstract — In this paper, we consider linear programming (LP) and second order cone programming (SOCP) based alternatives to sum of squares (SOS) programming and apply this framework to highdimensional problems arising in control applications. Despite the wide acceptance of SOS programming in the c ..."
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Abstract — In this paper, we consider linear programming (LP) and second order cone programming (SOCP) based alternatives to sum of squares (SOS) programming and apply this framework to highdimensional problems arising in control applications. Despite the wide acceptance of SOS programming in the control and optimization communities, scalability has been a key challenge due to its reliance on semidefinite programming (SDP) as its main computational engine. While SDPs have many appealing features, current SDP solvers do not approach the scalability or numerical maturity of LP and SOCP solvers. Our approach is based on the recent work of Ahmadi and Majumdar [1], which replaces the positive semidefiniteness constraint inherent in the SOS approach with stronger conditions based on diagonal dominance and scaled diagonal dominance. This leads to the DSOS and SDSOS cones of polynomials, which can be optimized over using LP and SOCP respectively. We demonstrate this approach on four high dimensional control problems that are currently well beyond the reach of SOS programming: computing a region of attraction for a 22 dimensional system, analysis of a 50 node network of oscillators, searching for degree 3 controllers and degree 8 Lyapunov functions for an Acrobot system (with the resulting controller validated on a hardware platform), and a balancing controller for a 30 state and 14 control input model of the ATLAS humanoid robot. While there is additional conservatism introduced by our approach, extensive numerical experiments on smaller instances of our problems demonstrate that this conservatism can be small compared to SOS programming. I.
Value Function Approximation Methods for Linearlysolvable Markov Decision Process
, 2013
"... Optimal control provides an appealing machinery to complete complicated control tasks with limited prior knowledge. Both global methods and online trajectory optimization methods are powerful techniques for solving optimal control problems; however, each has limitations. The global methods are direc ..."
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Optimal control provides an appealing machinery to complete complicated control tasks with limited prior knowledge. Both global methods and online trajectory optimization methods are powerful techniques for solving optimal control problems; however, each has limitations. The global methods are directly or indirectly based on the Bellman equation, which originates from dynamical programming. Finding the solution of Bellman equation, the value function, or costtogo function, suffers from multiple difficulties, including the curse of dimensionality. In the linearlysolvable Markov Decision Process (LMDP) framework, the Bellman equation can be linearized despite nonlinearity in the stochastic dynamical models. This fact permits efficient algorithms and motivates specialized function approximation schemes. In the averagecost setting, the Bellman equation in LMDP can be reduced to computing the principal eigenfunction of a linear operator. To solve for the value function of the Bellman equation in this cases, we designed two methods, moving least squares approximation and aggregation methods, to avoid matrix factorization and take advantage of sparsity by using efficient iterative solvers. In the moving least square approximation methods, value
Stability Analysis and Control of Rigid Body Systems with Impacts and Friction
"... Many critical tasks in robotics, such as locomotion or manipulation, involve collisions between a rigid body and the environment or between multiple bodies. Sumsofsquares (SOS) based methods for numerical computation of Lyapunov certificates are a powerful tool for analyzing the stability of cont ..."
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Many critical tasks in robotics, such as locomotion or manipulation, involve collisions between a rigid body and the environment or between multiple bodies. Sumsofsquares (SOS) based methods for numerical computation of Lyapunov certificates are a powerful tool for analyzing the stability of continuous nonlinear systems, and can additionally be used to automatically synthesize stabilizing feedback controllers. Here, we present a method for applying sumsofsquares verification to rigid bodies with Coulomb friction undergoing discontinuous, inelastic impact events. The proposed algorithm explicitly generates Lyapunov certificates for stability, positive invariance, and reachability over admissible (nonpenetrating) states and contact forces. We leverage the complementarity formulation of contact, which naturally generates the semialgebraic constraints that define this admissible region. The approach is demonstrated on multiple robotics examples, including simple models of a walking robot, a perching aircraft, and control design of a balancing robot.
Some Applications of Polynomial Optimization in Operations Research and RealTime Decision Making
"... We demonstrate applications of algebraic techniques that optimize and certify polynomial inequalities to problems of interest in the operations research and transportation engineering communities. Three problems are considered: (i) wireless coverage of targeted geographical regions with guaranteed ..."
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We demonstrate applications of algebraic techniques that optimize and certify polynomial inequalities to problems of interest in the operations research and transportation engineering communities. Three problems are considered: (i) wireless coverage of targeted geographical regions with guaranteed signal quality and minimum transmission power, (ii) computing realtime certificates of collision avoidance for a simple model of an unmanned vehicle (UV) navigating through a cluttered environment, and (iii) designing a nonlinear hovering controller for a quadrotor UV, which has recently been used for load transportation. On our smallerscale applications, we apply the sum of squares (SOS) relaxation and solve the underlying problems with semidefinite programming. On the largerscale or realtime applications, we use our recently introduced “SDSOS Optimization ” techniques which result in second order cone programs. To the best of our knowledge, this is the first study of realtime applications of sum of squares techniques in optimization and control. No knowledge in dynamics and control is assumed from the reader. 1
Synthesis and Optimization of Force Closure Grasps via Sequential Semidefinite Programming
"... Abstract In this paper we present a novel approach for synthesizing and optimizing both positions and forces in force closure grasps. This problem is a nonconvex optimization problem in general since it involves constraints that are bilinear; in particular, computing wrenches involves a bilinear ..."
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Abstract In this paper we present a novel approach for synthesizing and optimizing both positions and forces in force closure grasps. This problem is a nonconvex optimization problem in general since it involves constraints that are bilinear; in particular, computing wrenches involves a bilinear product between grasp contact points and contact forces. Thus, conventional approaches to this problem typically employ general purpose gradientbased nonlinear optimization. The key observation of this paper is that the force closure grasp synthesis problem can be posed as a Bilinear Matrix Inequality (BMI), for which there exist efficient solution techniques based on semidefinite programming. We show that we can synthesize force closure grasps on different geometric objects, and by maximizing a lower bound of a grasp metric, we can improve the quality of the grasp. While this approach is not guaranteed to find a solution, it has a few distinct advantages. First, we can handle nonsmooth but convex positive semidefinite constraints, which can often be important. Second, in contrast to gradientbased approaches we can prove infeasibility of problems. We demonstrate our method on a 15 joint robot model grasping objects with various geometries. The code is included in
Timedautomata abstraction of switched dynamical systems using control funnels?
"... Abstract. The development of formal methods for control design is an important challenge with potential applications in a wide range of safetycritical cyberphysical systems. Focusing on switched dynamical systems, we propose a new abstraction, based on timevarying regions of invariance (the contr ..."
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Abstract. The development of formal methods for control design is an important challenge with potential applications in a wide range of safetycritical cyberphysical systems. Focusing on switched dynamical systems, we propose a new abstraction, based on timevarying regions of invariance (the control funnels), that models behaviors of systems as timed automata. The main advantage of this method is that it allows automated verification of formal specifications and reactive controller synthesis without discretizing the evolution of the state of the system. Efficient constructions are possible in the case of linear dynamics. We demonstrate the potential of our approach with two examples. 1
Nonlinear Feedback Controllers via Occupation Measures
"... Abstract—In this paper, we present an approach for designing feedback controllers for polynomial systems that maximize the size of the timelimited backwards reachable set (BRS). We rely on the notion of occupation measures to pose the synthesis problem as an infinite dimensional linear program (LP) ..."
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Abstract—In this paper, we present an approach for designing feedback controllers for polynomial systems that maximize the size of the timelimited backwards reachable set (BRS). We rely on the notion of occupation measures to pose the synthesis problem as an infinite dimensional linear program (LP) and provide finite dimensional approximations of this LP in terms of semidefinite programs (SDPs). The solution to each SDP yields a polynomial control policy and an outer approximation of the largest achievable BRS. In contrast to traditional Lyapunov based approaches which are nonconvex and require feasible initialization, our approach is convex and does not require any form of initialization. The resulting timevarying controllers and approximated reachable sets are wellsuited for use in a trajectory library or feedback motion planning algorithm. We demonstrate the efficacy and scalability of our approach on five nonlinear systems. I.