Abstract
Signal temporal logic (STL) and reachability analysis are effective mathematical tools for formally analyzing the behavior of robotic systems. STL is a specification language that uses logic and temporal operators to precisely express real-valued and time-dependent requirements on system behaviors. While recursively defined STL specifications are extremely expressive and controller synthesis methods exist, there has not been work that quantifies the set of states from which STL formulas can be satisfied. Reachability analysis, on the other hand, involves computing the reachable set – the set of states from which a system is able to reach a goal while satisfying state and control constraints. While reasoning about system requirements through sets of states is useful for predetermining the possibility of satisfying desired system properties and obtaining state feedback controllers, so far the application of reachability has been limited to reach-avoid specifications. In this paper, we merge STL and time-varying reachability into a single framework that combines the key advantage of both methods – expressiveness of specifications and set quantification. To do this, we establish a correspondence between temporal and reachability operators, and use the idea of least-restrictive feasible controller sets (LRFCSs) to break down controller synthesis for complex STL formulas into a sequence of reachability and elementary set operations. LRFCSs are crucial for avoiding controller conflicts among different reachability operations. In addition, the synthesized state feedback controllers are guaranteed to satisfy STL specifications if determined to be possible by our framework, and violate specifications minimally if not. For simplicity, Hamilton-Jacobi reachability will be used in this paper, although our method is agnostic to the time-varying reachability method. We demonstrate our method through numerical simulations and robotic experiments.
M. Pavone—This work was supported by the Office of Naval Research YIP program (Grant N00014-17-1-2433) and by DARPA under the Assured Autonomy program. M. Chen and Q. Tam contributed equally to this work.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Some time-invariant reachability methods can be made time-varying by augmenting state space with time, as long as the resulting dynamics are still compatible with the reachability method.
- 2.
With a slight abuse of notation, we use \(\mathbb D\) to denote non-anticipative disturbance functions, intuitively control policies that do not depend on the future actions of another agent. See [28].
- 3.
The code used in this paper is available at https://github.com/StanfordASL/stlhj.
- 4.
- 5.
- 6.
- 7.
The videos can be viewed at
https://www.youtube.com/playlist?list=PL8-2mtIlFIJoNkhcGI7slWX2W3kEW-9Pb.
References
Synthesis of nonlinear continuous controllers for verifiably correct high-level, reactive behaviors. Int. J. Robot. Res
Althoff, M., Grebenyuk, D., Kochdumper, N.: Implementation of Taylor models in CORA 2018. In: Proceedings of the International Workshop on Applied Verification for Continuous and Hybrid Systems (2018)
Baier, C., Katoen, J.P.: Principles of Model Checking. MIT Press, Cambridge (2008)
Bansal, S., Chen, M., Fisac, J.F., Tomlin, C.J.: Safe sequential path planning of multi-vehicle systems under presence of disturbances and imperfect information. In: American Control Conference (2017)
Bansal, S., Chen, M., Herbert, S., Tomlin, C.J.: Hamilton-Jacobi reachability: a brief overview and recent advances. In: Proceedings of the IEEE Conference on Decision and Control (2017)
Belta, C., Bicchi, A., Egerstedt, M., Frazzoli, E., Klavins, E., Pappas, G.J.: Symbolic planning and control of robot motion [grand challenges of robotics]. IEEE Robot. Autom. Mag. 14(1), 61–70 (2007)
Chen, M., Herbert, S., Tomlin, C.J.: Fast reachable set approximations via state decoupling disturbances. In: Proceedings of the IEEE Conference on Decision and Control (2016)
Chen, M., Herbert, S.L., Vashishtha, M.S., Bansal, S., Tomlin, C.J.: Decomposition of reachable sets and tubes for a class of nonlinear systems. IEEE Trans. Autom. Control (2018, in press)
Chen, M., Hu, Q., Fisac, J., Akametalu, K., Mackin, C., Tomlin, C.: Reachability-based safety and goal satisfaction of unmanned aerial platoons on air highways. AIAA J. Guidance Control Dyn. 40(6), 1360–1373 (2017)
Chen, M., Tomlin, C.J.: Hamilton-Jacobi reachability: some recent theoretical advances and applications in unmanned airspace management. Annu. Rev. Control Robot. Auton. Syst. 1(1), 333–358 (2018)
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)
Donzé, A., Ferrère, T., Maler, O.: Efficient robust monitoring for STL. In: Proceedings of the International Conference Computer Aided Verification (2013)
Dreossi, T., Dang, T., Piazza, C.: Parallelotope bundles for polynomial reachability. In: Hybrid Systems: Computation and Control (2016)
Fainekos, G.E., Pappas, G.J.: Robustness of temporal logic specifications for continuous-time signals. Theor. Comput. Sci. 410(42), 4262–4291 (2009)
Feng, X., Villanueva, M.E., Chachuat, B., Houska, B.: Branch-and-lift algorithm for obstacle avoidance control. In: Proceedings of the IEEE Conference on Decision and Control (2017)
Fisac, J.F., Chen, M., Tomlin, C.J., Sastry, S.S.: Reach-avoid problems with time-varying dynamics, targets and constraints. In: Hybrid Systems: Computation and Control (2015)
Frehse, G., Le Guernic, C., Donzé, A., Cotton, S., Ray, R., Lebeltel, O., Ripado, R., Girard, A., Dang, T., Maler, O.: SpaceEx: scalable verification of hybrid systems. In: Proceedings of the International Conference Computer Aided Verification (2011)
Gattami, A., Al Alam, A., Johansson, K.H., Tomlin, C.J.: Establishing safety for heavy duty vehicle platooning: a game theoretical approach. IFAC World Congr. 44(1), 3818–3823 (2011)
Herbert, S.L., Chen, M., Han, S., Bansal, S., Fisac, J.F., Tomlin, C.J.: FaSTrack: a modular framework for fast and guaranteed safe motion planning. In: Proceedings of the IEEE Conference on Decision and Control (2017)
Kress-Gazit, H., Lahijanian, M., Raman, V.: Synthesis for robots: guarantees and feedback for robot behavior. Annu. Rev. Control Robot. Auton. Syst. 1, 211–236 (2018)
Landry, B., Chen, M., Hemley, S., Pavone, M.: Reach-avoid problems via sum-of-squares optimization and dynamic programming. In: IEEE/RSJ Intenational Conference on Intelligent Robots & Systems (2018)
Leung, K., Schmerling, E., Chen, M., Talbot, J., Gerdes, J.C., Pavone, M.: On infusing reachability-based safety assurance within probabilistic planning frameworks for human-robot vehicle interactions. In: International Symposium on Experimental Robotics (2018)
Majumdar, A., Tedrake, R.: Funnel libraries for real-time robust feedback motion planning (2016). https://arxiv.org/abs/1601.04037
Majumdar, A., Tedrake, R.: Funnel libraries for real-time robust feedback motion planning. Int. J. Robot. Res. 36(8), 947–982 (2017)
Majumdar, A., Vasudevan, R., Tobenkin, M.M., Tedrake, R.: Convex optimization of nonlinear feedback controllers via occupation measures. Int. J. Robot. Res. 33(9), 1209–1230 (2014)
Maler, O., Nickovic, D.: Monitoring temporal properties of continuous signals. In: Proceedings of the International Symposium Formal Techniques in Real-Time and Fault-Tolerant Systems, Formal Modeling and Analysis of Timed Systems (2004)
Mitchell, I.M.: The flexible, extensible and efficient toolbox of level set methods. SIAM J. Sci. Comput. 35(2–3), 300–329 (2008)
Mitchell, I.M., Bayen, A.M., Tomlin, C.J.: A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games. IEEE Trans. Autom. Control 50(7), 947–957 (2005)
Papusha, I., Fu, J., Topcu, U., Murray, R.M.: Automata theory meets approximate dynamic programming: optimal control with temporal logic constraints. In: Proceedings of the IEEE Conference on Decision and Control (2016)
Raman, V., Donze, A., Maasoumy, M., Murray, R.M., Sangiovanni-Vincentelli, A., Seshia, S.A.: Model predictive control with signal temporal logic specifications. In: Proceedings of the IEEE Conference on Decision and Control (2014)
Raman, V., Donzé, A., Sadigh, D., Murray, R.M., Seshia, S.A.: Reactive synthesis from signal temporal logic specifications. In: Hybrid Systems: Computation and Control (2015)
Castro, L.I.R., Chaudhari, P., Tumova, J., Karaman, S., Frazzoli, E., Rus, D.: Incremental sampling-based algorithm for minimum-violation motion planning. In: Proceedings of the IEEE Conference on Decision and Control (2013)
Singh, S., Majumdar, A., Slotine, J.J.E., Pavone, M.: Robust online motion planning via contraction theory and convex optimization. In: Proceedings of the IEEE Conference on Robotics and Automation (2017). http://asl.stanford.edu/wp-content/papercite-data/pdf/Singh.Majumdar.Slotine.Pavone.ICRA17.pdf
Wang, L., Ames, A.D., Egerstedt, M.: Safety barrier certificates for collisions-free multirobot systems. IEEE Trans. Robot. 33(3), 661–674 (2017). http://ieeexplore.ieee.org/document/7857061/
Wieland, P., Allgöwer, F.: Constructive safety using control barrier functions. In: IFAC Symposium on Nonlinear Control Systems (2007)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Chen, M., Tam, Q., Livingston, S.C., Pavone, M. (2020). Signal Temporal Logic Meets Reachability: Connections and Applications. In: Morales, M., Tapia, L., Sánchez-Ante, G., Hutchinson, S. (eds) Algorithmic Foundations of Robotics XIII. WAFR 2018. Springer Proceedings in Advanced Robotics, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-030-44051-0_34
Download citation
DOI: https://doi.org/10.1007/978-3-030-44051-0_34
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-44050-3
Online ISBN: 978-3-030-44051-0
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)