We present a compositional motion planning framework for multi-robot systems based on an encoding to satisfiability modulo theories (SMT). In our framework, the desired behavior of a group of robots is specified using a set of safe linear temporal logic (LTL) properties. Our method relies on a library of motion primitives, each of which corresponds to a controller that ensures a particular trajectory in a given configuration. Using the closed-loop behavior of the robots under the action of different controllers, we formulate the motion planning problem as an SMT solving problem and use an off-the-shelf SMT solver to generate trajectories for the robots. Our approach can also be extended to synthesize optimal cost trajectories where optimality is defined with respect to the available motion primitives. Experimental results show that our framework can efficiently solve complex motion planning problems in the context of multi-robot systems.
A key feature of control systems is robustness, the property that small perturbations in the system inputs cause only small changes in its outputs. Robustness is key to designing systems that work under uncertain or imprecise environments. While continuous control design algorithms can explicitly incorporate robustness as a design goal, it is not clear if robustness is maintained at the software implementation level of the controller: two "close" inputs can execute very different code paths which may potentially produce vastly different outputs.We present an algorithm and a tool to characterize the robustness of a control software implementation. Our algorithm is based on symbolic execution and non-linear optimization, and computes the maximum difference in program outputs over all program paths when a program input is perturbed. As a by-product, our algorithm generates a set of test vectors which demonstrate the worst-case deviations in outputs for small deviations in inputs. We have implemented our approach on top of the Splat test generation tool and we describe an evaluation of our implementation on two examples of automotive control code.
Several problems in the implementations of control systems, signal-processing systems, and scientific computing systems reduce to compiling a polynomial expression over the reals into an imperative program using fixed-point arithmetic. Fixed-point arithmetic only approximates real values, and its operators do not have the fundamental properties of real arithmetic, such as associativity. Consequently, a naive compilation process can yield a program that significantly deviates from the real polynomial, whereas a different order of evaluation can result in a program that is close to the real value on all inputs in its domain.We present a compilation scheme for real-valued arithmetic expressions to fixed-point arithmetic programs. Given a real-valued polynomial expression t, we find an expression t that is equivalent to t over the reals, but whose implementation as a series of fixed-point operations minimizes the error between the fixed-point value and the value of t over the space of all inputs. We show that the corresponding decision problem, checking whether there is an implementation t of t whose error is less than a given constant, is NP-hard. We then propose a solution technique based on genetic programming. Our technique evaluates the fitness of each candidate program using a static analysis based on affine arithmetic. We show that our tool can significantly reduce the error in the fixed-point implementation on a set of linear control system benchmarks. For example, our tool found implementations whose errors are only one half of the errors in the original fixed-point expressions.
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