Local stability analysis using simulations and sum-of-squares programming

Topcu, Ufuk; Packard, Andrew; Seiler, Peter

doi:10.1016/j.automatica.2008.03.010

Cited by 207 publications

(219 citation statements)

References 17 publications

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“…The computational algorithm is briefly described here and full algorithmic details are provided elsewhere (Jarvis-Wloszek, 2003;Jarvis-Wloszek et al, 2003;Tan and Packard, 2004;Jarvis-Wloszek et al, 2005;Tan, 2006;Topcu et al, 2007Topcu et al, , 2008. Lemma 1 is the main Lyapunov theorem used to compute lower bounds on β * .…”

Section: Region Of Attraction Estimationmentioning

confidence: 99%

“…This nonlinear short period model is of interest because the decoupling of the longitudinal modes is typically done using linearized models. Section 3 describes a computational procedure to estimate regions of attraction for polynomial systems (Jarvis-Wloszek, 2003;JarvisWloszek et al, 2003;Tan and Packard, 2004;Jarvis-Wloszek et al, 2005;Tan, 2006;Topcu et al, 2007Topcu et al, , 2008. This algorithm is applied in Section 4 to estimate regions of attractions for the open-loop short period dynamics and a closed-loop longitudinal GTM aircraft.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, significant research has been performed on the development of nonlinear analysis tools for computing regions of attraction, reachability sets, input-output gains, and robustness with respect to uncertainty for nonlinear polynomial systems (Tan, 2006;Tan et al, 2008;Topcu et al, 2007Topcu et al, , 2008Chiang and Thorp, 1989;Davison and Kurak, 1971;Genesio et al, 1985;Tibken, 2000;Tibken and Fan, 2006;Vannelli and Vidyasagar, 1985;Parrilo, 2000). These tools make use of polynomial sum-of-squares optimization (Parrilo, 2000).…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear region of attraction analysis for flight control verification and validation

Chakraborty

Seiler

Balas

2011

Control Engineering Practice

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Current practice for flight control validation relies heavily on linear analyses and nonlinear, high-fidelity simulations. This process would be enhanced by the addition of nonlinear analyses of the flight control system. This paper demonstrates the use of region of attraction estimation for studying nonlinear effects. A nonlinear polynomial model is constructed for the longitudinal dynamics of NASA's Generic Transport Model aircraft. A polynomial model for the short period dynamics is obtained by decoupling this mode from the nonlinear longitudinal model. Polynomial optimization techniques are applied to estimate region of attractions around trim conditions. CONTROL ENGINEERING PRACTICEPAPER CHECKLIST for the preparation of papers.Please ensure that your paper conforms to the following guidelines. PREFERRED FORMAT AND LAYOUT:

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Section: Region Of Attraction Estimationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonlinear region of attraction analysis for flight control verification and validation

Chakraborty

Seiler

Balas

2011

Control Engineering Practice

Self Cite

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“…With the Sprocedure multipliers, the above optimization program will be bilinear in the unknown polynomials (V appears both in the antecedent and consequent of the implications). To solve it, we adopt a two-stage technique of bilinear alternations, similar to the approaches used in [37,21,28]. While these approaches offer no guarantee of optimality, they are practically effective and relatively straightforward to implement.…”

Section: A Bilinear Inner Approximationmentioning

confidence: 99%

Balancing and Step Recovery Capturability via Sums-of-Squares Optimization

Posa

Koolen

Tedrake

2017

Robotics: Science and Systems XIII

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Abstract-A fundamental requirement for legged robots is to maintain balance and prevent potentially damaging falls whenever possible. As a response to outside disturbances, fall prevention can be achieved by a combination of active balancing actions, e.g. through ankle torques and upper-body motion, and through reactive step placement. While it is widely accepted that stepping is required to respond to large disturbances, the limits of active motions on balancing and step recovery are only well understood for the simplest of walking models. Recent advances in convex optimization-based verification and control techniques enable a more complete understanding of the limits and capabilities of more complex models. In this work, we present an algorithmic approach for formal analysis of the viable-capture basins of walking robots, calculating both inner and outer approximations and corresponding push recovery control strategies. Extending beyond the classic Linear Inverted Pendulum Model (LIPM), we analyze a series of centroidal momentum based planar walking models, examining the effects of center of mass height, angular momentum, and impact dynamics during stepping on capturability. This formal analysis enables an explicit calculation of the differences between these models, and assessment of whether the simplest models ultimately sacrifice capability, and thus stability, when designing push recovery control policies.

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“…In general, SOS programming has had a large impact on the control theory community since its advent over a decade ago [80] and has been used to tackle a wide variety of problems including feedback control synthesis, safety verification and computation of regions of attraction, invariant sets, and reachable sets for a broad class of nonlinear and hybrid systems [105,49,87,28,2,46]. However, despite the wide acceptance of the SOS approach in the control and optimization communities, applications of the method considered in the literature typically involve systems of relatively modest dimension (approximately 5-10 states).…”

Section: Chaptermentioning

confidence: 99%

Funnel libraries for real-time robust feedback motion planning

Majumdar

Tedrake

2017

The International Journal of Robotics Research

310

270

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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. 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, the resulting hardware demonstrations on a fixed-wing airplane constitute one of the first examples of provably safe and robust control for robotic systems with complex nonlinear dynamics that need to plan in realtime in environments with complex geometric constraints.The key computational engine we leverage is sums-of-squares (SOS) programming. While SOS programming allows us to apply our approach to systems of relatively high dimensionality (up to approximately 10-15 dimensional state spaces), scaling our approach to higher dimensional systems such as humanoid robots requires a different set of computational tools. In this thesis, we demonstrate how DSOS and SDSOS programming, which are recently introduced alternatives to SOS programming, can be employed to achieve this improved scalability and handle control systems with as many as 30-50 state dimensions. My conversations with him on measure theory, functional analysis, and algebraic topology were extremely enriching and provided me with a set of technical tools that will no doubt be very valuable going forwards. I am grateful to Hongkai for giving me the chance to explore problems in grasping and manipulation with him. Hongkai's incredible capacity to work long hours have been an inspiration throughout my time as a graduate student and have pushed me to work that extra bit longer myself. I have lost count of the number of times a quick late night conversation with him has helped solve a problem that I was stuck on.I would also like to thank the other members of the Robot Locomotion Group for inspiration, advice, help, and entertainment during my time here. It has been an honor to work with people who will no doubt lead our field in years to come. Ian Manchester, Zack Jackowski, Michael Levashov, John Roberts,...

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Local stability analysis using simulations and sum-of-squares programming

Cited by 207 publications

References 17 publications

Nonlinear region of attraction analysis for flight control verification and validation

Nonlinear region of attraction analysis for flight control verification and validation

Balancing and Step Recovery Capturability via Sums-of-Squares Optimization

Funnel libraries for real-time robust feedback motion planning

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