125 lines
5.6 KiB
Markdown
125 lines
5.6 KiB
Markdown
---
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title: Path Planning With TWTLPlan
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summary: Sampling-based path planning using temporal logic specifications
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tags: []
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date: "2021-02-03"
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# Optional external URL for project (replaces project detail page).
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external_link: ""
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image:
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caption: ""
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focal_point: Smart
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links:
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# - icon: twitter
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# icon_pack: fab
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# name: Follow
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# url: https://twitter.com/georgecushen
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url_code: "https://github.com/fran-penedo/twtlplan"
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url_pdf: ""
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url_slides: ""
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url_video: ""
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# Slides (optional).
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# Associate this project with Markdown slides.
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# Simply enter your slide deck's filename without extension.
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# E.g. `slides = "example-slides"` references `content/slides/example-slides.md`.
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# Otherwise, set `slides = ""`.
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slides: ""
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---
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Motion planning is a fundamental problem in robotics. The
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objective is to generate control policies for a robot to drive it from
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an initial state to a goal region in its state space under
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kino-dynamic constraints
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Even without considering dynamics, the problem becomes increasingly
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difficult in high dimensions, and it has been shown
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to be PSPACE-complete. A class of algorithms
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was developed relying on randomly sampling
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the configuration space of the robot and planning local
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motions between these samples.
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Probabilistic Roadmaps and Rapidly Exploring
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Random Trees are among the most widely known
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examples, along with their asymptotically optimal variants,
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PRM$^\*$ and RRT$^\*$.
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Robots are increasingly required to perform complex tasks,
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where correctness guarantees, such as safety and liveness
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in human-robot teams and autonomous driving, are critical.
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One approach is to encode the tasks as temporal logic specifications
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and leverage formal methods techniques to generate
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control policies that are correct by construction.
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As opposed to traditional methods restricted to reach-avoid setups,
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these frameworks are able to express more complex
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tasks such as sequencing (e.g., "Reach $A$, then $B$"),
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convergence ("Go to $A$ and stay there forever"),
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persistent surveillance ("Visit $A$, $B$, and $C$ infinitely often"),
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and more complex combinations of the above.
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Some applications additionally require time constraints.
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For example, consider the following task:
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"Visit $A$, $B$, and $C$ in this order. Perform action $a$ for 2 time units at
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$A$ within 10 time units. Then, perform $b$ for 3 time
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units at $B$ within 6 time units. Finally, in the time window $[3, 9]$ after $b$ is finished,
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perform $c$ for 1 time unit at $C$. All three actions must be finished
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within 15 time units."
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Temporal logics such as Linear Temporal Logic (LTL) or Computational Tree Logic
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(CTL) can be used to encode the former kind of tasks, while tasks with explicit time
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constraints may be expressed using bounded linear temporal
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logic (BLTL), metric temporal logic (MTL),
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signal temporal logic (STL), and time-window temporal logic
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(TWTL).
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A natural approach to generate control strategies from
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rich task specifications for robots with large state spaces is to combine
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sampling-based techniques with automata-based synthesis methods.
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In this project, we proposed a language-guided sampling-based method to generate
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robot control policies satisfying tasks expressed as TWTL formulae.
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A TWTL formula
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formally captures the notion of performing tasks of a certain duration in regions of interest
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in order. For example, a specification in natural language such as
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"perform task A of duration 1 within 2 time units; then, within the time
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interval [1, 8] perform tasks B and C of durations 3 and 2, respectively; furthermore, C
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must be finished within 4 time units from the start of B;" would correspond to the
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following TWTL formula:
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$$
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f = [H^1 A]^{[0, 2]} * [H^3 B \wedge [H^2 C]^{[0, 4]}]^{[1, 8]}
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$$
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A key feature of TWTL is that a formula may be relaxed, i.e., we can add a variable
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delay to each deadline. A relaxed formula with a positive delay is a weaker
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specification, while a negative delay indicates a stronger specification. A path
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satisfying a stronger specification satisfies a weaker one. TWTLPlan makes use of this
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feature by finding a path that satisfies a weaker specification first, then attempts to
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reduce the delay until the original specification is satified. These paths are computed
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using a sampling-based algorithm called RRT$^\*$.
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This approach has two advantages:
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(a) the growth of the sampling graph is biased towards satisfaction of a
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relaxed version of the specification
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without initially taking into account deadlines, and
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(b) after an initial policy has been found, sampling can be focused on the parts
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of the plan that need to be improved.
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The two stages can be thought of as exploration-exploitation phases,
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where initially candidate solutions are found,
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and then focused local sampling is performed on the parts
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that need to be improved in order to satisfy the specification, i.e. time
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constraints.
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As a byproduct, a satisfying policy with respect to a minimally
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relaxed version of the specification may be returned
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when the original problem does not have a solution.
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Such a policy may inform operators about timing problems in the specification
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or system (i.e., robot dynamics and/or environment).
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The algorithm uses annotated finite state automata to
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represent all possible temporal relaxations of a TWTL formulae.
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We were also able to prove that our solution is probabilistically complete.
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We implemented our path planning algorithm in [TWTLPlan](https://github.com/fran-penedo/twtlplan).
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Using this tool you can define 2D scenarios with rectangular obstacles and regions of
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interest and generate satisfying paths for specifications given in a subset of TWTL.
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