2.3: Piano Mover’s Problem

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From the course by University of Pennsylvania

Robotics: Computational Motion Planning

712 ratings

University of Pennsylvania

Robotics: Computational Motion Planning

712 ratings

Course 2 of 6 in the Specialization Robotics

Robotic systems typically include three components: a mechanism which is capable of exerting forces and torques on the environment, a perception system for sensing the world and a decision and control system which modulates the robot's behavior to achieve the desired ends. In this course we will consider the problem of how a robot decides what to do to achieve its goals. This problem is often referred to as Motion Planning and it has been formulated in various ways to model different situations. You will learn some of the most common approaches to addressing this problem including graph-based methods, randomized planners and artificial potential fields. Throughout the course, we will discuss the aspects of the problem that make planning challenging.

From the lesson

Configuration Space

Welcome to Week 2! In this module, we begin by introducing the concept of configuration space which is a mathematical tool that we use to think about the set of positions that our robot can attain. We then discuss the notion of configuration space obstacles which are regions in configuration space that the robot cannot take on because of obstacles or other impediments. This formulation allows us to think about path planning problems in terms of constructing trajectories for a point through configuration space. We also describe a few approaches that can be used to discretize the continuous configuration space into graphs so that we can apply graph-based tools to solve our motion planning problems.

2.1: Introduction to Configuration Space4:01

2.2: RR arm2:17

2.3: Piano Mover’s Problem3:03

Meet the Instructors

CJ Taylor
Professor of Computer and Information Science
School of Engineering and Applied Science

Here's a more complicated example, once again,

we are considering a robot that can translate in the plane.

But now,

it can also rotate, this means our robot now has three degrees of freedom.

Since a rotational degree of freedom has been added

to its initial two translational degrees.

We can denote the configuration of our robot with a tuple, tx, ty, and theta,

where tx and ty still denote the position of a reference point in the plane, and

theta denotes the applied rotational angle in degrees.

Once again, when we introduce obstacles into the workspace,

we can think about the set of configurations that are limited.

In this case, the configuration space has three dimensions, and the configuration

space obstacles can be thought of as three dimensional regions in this space.

This movie shows a depiction of the surface of the configuration space

obstacle corresponding to the obstacles shown in the previous figure.

The vertical access corresponds to the rotation theta, while the other

two horizontal axes correspond to the translational parameters tx and

ty Note again that in this figure, the surface that we

are visualizing corresponds to the surface of the configuration space obstacle.

As before, the basic problem in motion planning is to come up with a trajectory

between a start point and

an end point that avoids all the configuration space obstacles.

This movie shows a robot moving through the space, avoiding all of the obstacles.

In this second movie, we are visualizing the trajectory of

the robot through configuration space as a red line.

Notice how this red line snakes in and

around the configuration space obstacle avoiding penetration as it moves

from the start configuration to the end of configuration.

It is important to understand that this idea of a configuration space where we

associate coordinates with the configuration of the robot and

then reason about configurations that are allowed and disallowed, and

think about the motion of the robot in terms of trajectories of a point through

configuration space is actually very general.

Here, for example, is a plane a robot with six revolute links

In principle, we can think of its motion in terms of its trajectory of a point,

moving through a six dimensional configuration space.

If we wanted to, we could introduce obstacles in the space and

reason about the corresponding configuration obstacles.

I invite you to ponder what this configuration space would

actually look like.

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