| dc.description.abstract | Simultaneous localization and planning for nonlinear stochastic systems under
process and measurement uncertainties is a challenging problem. In its most general
form, it is formulated as a stochastic optimal control problem in the space of feedback
policies. The Hamilton-Jacobi-Bellman equation provides the theoretical solution of
the optimal problem; but, as is typical of almost all nonlinear stochastic systems,
optimally solving the problem is intractable. Moreover, even if an optimal solution
was obtained, it would require centralized control, while multi-agent mobile robotic
systems under dynamic environments require decentralized solutions.
In this study, we aim for a theoretically sound solution for various modes of
this problem, including the single-agent and multi-agent variations with perfect and
imperfect state information, where the underlying state, control and observation
spaces are continuous with discrete-time models. We introduce a decoupling principle
for planning and control of multi-agent nonlinear stochastic systems based on a
small noise asymptotics. Through this decoupling principle, under small noise, the
design of the real-time feedback law can be decoupled from the off-line design of the
nominal trajectory of the system. Further, for a multi-agent problem, the design of
the feedback laws for different agents can be decoupled from each other, reducing the
centralized problem to a decentralized problem requiring no communication during
execution. The resulting solution is quantifiably near-optimal.
We establish this result for all the above-mentioned variations, which results in
the following variants: Trajectory-optimized Linear Quadratic Regulator (T-LQR),
Multi-agent T-LQR (MT-LQR), Trajectory-optimized Linear Quadratic Gaussian
(T-LQG), and Multi-agent T-LQG (MT-LQG). The decoupling principle provides the conditions under which a decentralized linear Gaussian system with a quadratic
approximation of the cost, obtained by linearization around an optimally designed
nominal trajectory can be utilized to control the nonlinear system. The resulting decentralized
feedback solution at runtime, being decoupled with respect to the mobile
agents, requires no communication between the agents during the execution phase.
Moreover, the complexity of the solution vis-a-vis the computation of the nominal
trajectory as well as the closed-loop gains is tractable with low polynomial orders of
computation. Experimental implementation of the solution shows that the results
hold for moderate levels of noise with high probability.
Further optimizing the performance of this approach we show how to design a
special cost function for the problem with imperfect state measurement that takes
advantage of the fact that the estimation covariance of a linear Gaussian system is
deterministic and not dependent on the observations. This design, which corresponds
in our overall design to “belief space planning”, incorporates the consequently deterministic
cost of the stochastic feedback system into the deterministic design of the
nominal trajectory to obtain an optimal nominal trajectory with the best estimation
performance. Then, it utilizes the T-LQG approach to design an optimal feedback
law to track the designed nominal trajectory. This iterative approach can be used to
further tune both the open loop as well as the decentralized feedback gain portions
of the overall design. We also provide the multi-agent variant of this approach based
on the MT-LQG method.
Based on the near-optimality guarantees of the decoupling principle and the TLQG
approach, we analyze the performance and correctness of a well-known heuristic
in robotic path planning. We show that optimizing measures of the observability
Gramian as a surrogate for estimation performance may provide irrelevant or misleading
trajectories for planning under observation uncertainty.
We then consider systems with non-Gaussian perturbations. An alternative
heuristic method is proposed that aims for fast planning in belief space under non-
Gaussian uncertainty. We provide a special design approach based on particle filters
that results in a convex planning problem implemented via a model predictive control
strategy in convex environments, and a locally convex problem in non-convex environments.
The environment here refers to the complement of the region in Euclidean
space that contains the obstacles or “no fly zones”.
For non-convex dynamic environments, where the no-go regions change dynamically
with time, we design a special form of an obstacle penalty function that incorporates
non-convex time-varying constraints into the cost function, so that the
decoupling principle still applies to these problems. However, similar to any constrained
problem, the quality of the optimal nominal trajectory is dependent on the
quality of the solution obtainable for the nonlinear optimization problem.
We simulate our algorithms for each of the problems on various challenging situations,
including for several nonlinear robotic models and common measurement
models. In particular, we consider 2D and 3D dynamic environments for heterogeneous
holonomic and non-holonomic robots, and range and bearing sensing models.
Future research can potentially extend the results to more general situations including
continuous-time models. | en |
