Optimal Control of Polynomial Hybrid Systems via Convex Relaxations

Zhao, Pengcheng; Mohan, Shankar; Vasudevan, Ram

doi:10.1109/tac.2019.2929110

Cited by 20 publications

(33 citation statements)

References 50 publications

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“…We transform each (D i ) into a semi-definite program (SDP) using SOS programming via the Spotless toolbox [24], as covered in detail by [22], [23], [25]. We solve the SDP with MOSEK [26].…”

Section: Methodsmentioning

confidence: 99%

“…We solve the SDP with MOSEK [26]. The key implementation difference is, where [23] and [25] solve a single SDP over multiple hybrid system modes, we solve a sequence of SDPs for each phase T i , with i ∈ {move,brake,stop} and the initial condition sets X i,0 implemented as discussed above. For each i, the sequence of SDPs return (v i , w i , q i ) as polynomials of fixed degree.…”

Section: Methodsmentioning

confidence: 99%

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Towards Provably Not-At-Fault Control of Autonomous Robots in Arbitrary Dynamic Environments

Vaskov¹,

Kousik²,

Larson³

et al. 2019

Robotics: Science and Systems XV

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As autonomous robots increasingly become part of daily life, they will often encounter dynamic environments while only having limited information about their surroundings. Unfortunately, due to the possible presence of malicious dynamic actors, it is infeasible to develop an algorithm that can guarantee collision-free operation. Instead, one can attempt to design a control technique that guarantees the robot is not-at-fault in any collision. In the literature, making such guarantees in real time has been restricted to static environments or specific dynamic models. To ensure not-at-fault behavior, a robot must first correctly sense and predict the world around it within some sufficiently large sensor horizon (the prediction problem), then correctly control relative to the predictions (the control problem). This paper addresses the control problem by proposing Reachability-based Trajectory Design for Dynamic environments (RTD-D), which guarantees that a robot with an arbitrary nonlinear dynamic model correctly responds to predictions in arbitrary dynamic environments. RTD-D first computes a Forward Reachable Set (FRS) offline of the robot tracking parameterized desired trajectories that include fail-safe maneuvers. Then, for online receding-horizon planning, the method provides a way to discretize predictions of an arbitrary dynamic environment to enable real-time collision checking. The FRS is used to map these discretized predictions to trajectories that the robot can track while provably not-at-fault. One such trajectory is chosen at each iteration, or the robot executes the fail-safe maneuver from its previous trajectory which is guaranteed to be not at fault. RTD-D is shown to produce not-at-fault behavior over thousands of simulations and several real-world hardware demonstrations on two robots: a Segway, and a small electric vehicle.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Towards Provably Not-At-Fault Control of Autonomous Robots in Arbitrary Dynamic Environments

Vaskov¹,

Kousik²,

Larson³

et al. 2019

Robotics: Science and Systems XV

Self Cite

View full text Add to dashboard Cite

show abstract

“…However, the representation of each of these sets in state space severely restricts the size of the problem that can be tackled by these approaches. To accommodate this limitation, sums-of-squares analysis has been primarily applied to reduced models of walking robots: ranging from spring mass models [13], to inverted pendulum models [10], [14] and to inverted pendulum models with an offset torso mass [12]. The substantial differences between these simple models and real robots causes difficulty when applying these results to hardware.…”

Section: Introductionmentioning

confidence: 99%

Walking With Confidence: Safety Regulation for Full Order Biped Models

Smit-Anseeuw

Remy

Vasudevan

2019

IEEE Robot. Autom. Lett.

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Safety guarantees are valuable in the control of walking robots, as falling can be both dangerous and costly. Unfortunately, set-based tools for generating safety guarantees (such as sums-of-squares optimization) are typically restricted to simplified, low-dimensional models of walking robots. For more complex models, methods based on hybrid zero dynamics can ensure the local stability of a pre-specified limit cycle, but provide limited guarantees. This paper combines the benefits of both approaches by using sums-of-squares optimization on a hybrid zero dynamics manifold to generate a guaranteed safe set for a 10-dimensional walking robot model. Along with this set, this paper describes how to generate a controller that maintains safety by modifying the manifold parameters when on the edge of the safe set. The proposed approach, which is applied to a bipedal Rabbit model, provides a roadmap for applying sums-of-squares techniques to high dimensional systems. This opens the door for a broad set of tools that can generate flexible and safe controllers for complex walking robot models.

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“…It has been applied to the approximation of the region of attraction, the backward reachable set, and the maximum controllable set for continuous-time polynomial systems [8]- [10], [21]. It has also been applied to controller synthesis for continuous-time nonhybrid/hybrid polynomial systems [9], [14], [25].…”

Section: Introductionmentioning

confidence: 99%

Controller Synthesis for Discrete-Time Polynomial Systems via Occupation Measures

Han

Tedrake

2018

2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)

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In this paper, we design nonlinear state feedback controllers for discrete-time polynomial dynamical systems via the occupation measure approach. We propose the discrete-time controlled Liouville equation, and use it to formulate the controller synthesis problem as an infinite-dimensional linear programming problem on measures, which is then relaxed as finitedimensional semidefinite programming problems on moments of measures and their duals on sums-of-squares polynomials. Nonlinear controllers can be extracted from the solutions to the relaxed problems. The advantage of the occupation measure approach is that we solve convex problems instead of generally non-convex problems, and the computational complexity is polynomial in the state and input dimensions, and hence the approach is more scalable. In addition, we show that the approach can be applied to over-approximating the backward reachable set of discrete-time autonomous polynomial systems and the controllable set of discrete-time polynomial systems under known state feedback control laws. We illustrate our approach on several dynamical systems.

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Optimal Control of Polynomial Hybrid Systems via Convex Relaxations

Cited by 20 publications

References 50 publications

Towards Provably Not-At-Fault Control of Autonomous Robots in Arbitrary Dynamic Environments

Towards Provably Not-At-Fault Control of Autonomous Robots in Arbitrary Dynamic Environments

Walking With Confidence: Safety Regulation for Full Order Biped Models

Controller Synthesis for Discrete-Time Polynomial Systems via Occupation Measures

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