What is Configuration Space in Robotics? Best Explanation

The configuration of a robot system, which is a specification of the position 

of every point of the robot. Since the robot consists of a collection of rigid 

bodies connected by joints, our study begins with understanding the configuration 

of a rigid body.

We see that the configuration of a rigid body in the plane can be described

using three variables (two for the position and one for the orientation) and the

configuration of a rigid body in space can be described using six variables (three

for the position and three for the orientation). The number of variables is the

number of degrees of freedom (dof) of the rigid body. It is also the dimension

of the configuration space, the space of all configurations of the body.

The dof of a robot, and hence the dimension of its configuration space, is

the sum of the dof of its rigid bodies minus the number of constraints on the

motion of those rigid bodies provided by the joints. 

For example, the two most

popular joints, revolute (rotational) and prismatic (translational) joints, allow

only one motion freedom between the two bodies they connect. Therefore a

revolute or prismatic joint can be thought of as providing five constraints on

the motion of one spatial rigid body relative to another. Knowing the dof of

a rigid body and the number of constraints provided by joints, we can derive

Gr¨ubler’s formula for calculating the dof of general robot mechanisms. For

open-chain robots such as the industrial manipulator of Figure 1.1(a), each

joint is independently actuated and the dof is simply the sum of the freedoms

provided by each joint. 

For closed chains like the Stewart–Gough platform

in Figure 1.1(b), Gr¨ubler’s formula is a convenient way to calculate a lower

bound on the dof. Unlike open-chain robots, some joints of closed chains are

not actuated. Apart from calculating the dof, other configuration space concepts 

of interest include the topology (or “shape”) of the configuration space and its 

representation. Two configuration spaces of the same dimension may have different

shapes, just like a two-dimensional plane has a different shape from the two 

dimensional surface of a sphere. These differences become important when 

determining how to represent the space. The surface of a unit sphere, 

for example, could be represented using a minimal number of coordinates, 

such as latitude and longitude, or it could be represented by three numbers (x, y, z) 

subject to the constraint x2 + y2 + z2 = 1. The former is an explicit parametrization

of the space and the latter is an implicit parametrization of the space. Each

type of representation has its advantages, but in this book we will use implicit

representations of configurations of rigid bodies.

A robot arm is typically equipped with a hand or gripper, more generally

called an end-effector, which interacts with objects in the surrounding world.

To accomplish a task such as picking up an object, we are concerned with the

configuration of a reference frame rigidly attached to the end-effector, and not

necessarily the configuration of the entire arm. We call the space of positions

and orientations of the end-effector frame the task space and note that there

is not a one-to-one mapping between the robot’s configuration space and the

task space. The workspace is defined to be the subset of the task space that

the end-effector frame can reach.