Motion Planning with Uncertainty

Sources of Uncertainty

Incompleteness of world model: E.g., one or several obstacles are not part of the robot's model of the world.

E.g.: Mobile robot entering a new environment.
Unaccurate world model: E.g., the location of an obstacle is known within some tolerances.

E.g.: Mobile robot navigating in a familiar environment.
Uncertainty in sensing: Sensed data are not accurate.

E.g.: Mobile robot localization, part mating in assembly.
Uncertainty in control: The robot does not exactly follow a commanded path.

E.g.: Mobile robot navigation, manipulator arm performing operations requiring high precision.

Motion Planning with Incomplete Knowledge of Workspace

Approaches:

Track obstacles' boundaries, Bug algorithms of [Lumelsky].

Reference:
Lumelsky. Algorithmic Issues of Sensor-Based Robot Motion Planning. Proc. 26th Conf. on Decision and Control Los Angeles, CA, pp. 1796-1801.
Track steepest descent of a potential field with a repulsive component derived from sensed obstacles [Khatib].
Iterate sense/plan/execute [Chatila]

Motion Planning with Uncertainty in Control and Sensing

Motion command

Assume the robot is a point in the plane.

A motion command M consists of two parts:

The control part, which specifies how the robot should move, e.g., a commanded velocity v.
The termination condition, which specifies when the motion should stop. The termination condition is a boolean function (predicate) of the elapsed time and the sensing history since the beginning of the motion.

The execution of the command M leads the robot to move with a velocity v' that is, at each time, slightly different of v (uncertainty in control), both in orientation and in modulus.

Furthermore, during motion, sensing is imperfect. For example, at any one time, the actual and measured positions of the robot may differ slightly.

Motion planning problem

Given:

An initial region I in which the robot is known to be prior to executing a motion plan,
A goal region G in which we want the robot to be when the motion plan has been executed,
Bounds on the control ans sensing errors,

Compute:

A motion plan (e.g., a sequence of motion commands) whose execution guarantees that the robot starting from anywhere withing I will stop in G (guaranteed plan).

A word on Terminology

Error: Difference between actual and commanded of sensed value.

Uncertainty: Distribution of errors.

We will assume that all errors are uniformly distributed over a closed and bounded domain (e.g., an interval, a disc).

Subproblems of Planning with Uncertainty in Control and Sensing

Identify a set of states relevant to the problem.
Select motion commands to transit from states to states.
Design termination conditions to recognize achievement of states.

All three subproblems interact. They cannot be solved in isolation.

Notion of a Preimage

The preimage of a goal G for a motion command M is a subset P (ideally the largest) such that executing M from anywhere in P guarantees that the robot will stop in G.

References:
Lozano-Perez, Mason, and Taylor. Automatic Synthesis of Fine-Motion Strategies for Robots. Int'l J. of Robotics Research, 3(1), 3-24, 1984.

Preimage Backchaining

Planning technique that consists of computing preimages of the goal region G and, recursively, preimages of previously computed preimages, until one preimage contains the initial region I.

Preimage backchaining raises the following computational isues:

Computation of maximal preimages for given commanded velocities.
Choice of commanded velocities.
Generation of conditional plans.

Termination Condition

The size of a preimage depends on whether the termination condition uses:

Elapsed time, or not.
Sensing history, or not.

and on whether its definition embeds knowledge of the initial state (preimage itself) and final state.

Simple problem

(just to check your understanding!)

A robot represented by a point moves along a line, with imperfect velocity control and imperfect position sensing.

Since the robot moves along a line, there is no uncertainty in the direction of the velocity, but there is uncertainty e in the velocity modulus, i.e., if |v| is the modulus of the commanded velocity, the modulus of the actual velocity is at any time contained in the interval [|v|-e,|v|+e].

The uncertainty on position sensing is d, i.e., if the robot senses that it is at abscissa x along the line, it may actually be anywhere in [|x|-d,|x|+d].

The goal is defined to be the interval [3,4].

Is the half-line x <= 4 a preimage of this goal? If yes, what is the termination condition?

Answer

Preimage Computation: Kernel/Backprojection Approach

Principle: Break the dependence between the preimage and the termination condition.

But ... the computed preimage is not the largest in general.

Two steps in the computation