  The signed distance field is a popular implicit shape representation in
robotics, providing geometric information about objects and obstacles in a form
that can easily be combined with control, optimization and learning techniques.
Most often, SDFs are used to represent distances in task space, which
corresponds to the familiar notion of distances that we perceive in our 3D
world. However, SDFs can mathematically be used in other spaces, including
robot configuration spaces. For a robot manipulator, this configuration space
typically corresponds to the joint angles for each articulation of the robot.
While it is customary in robot planning to express which portions of the
configuration space are free from collision with obstacles, it is less common
to think of this information as a distance field in the configuration space. In
this paper, we demonstrate the potential of considering SDFs in the robot
configuration space for optimization, which we call the configuration space
distance field. Similarly to the use of SDF in task space, CDF provides an
efficient joint angle distance query and direct access to the derivatives. Most
approaches split the overall computation with one part in task space followed
by one part in configuration space. Instead, CDF allows the implicit structure
to be leveraged by control, optimization, and learning problems in a unified
manner. In particular, we propose an efficient algorithm to compute and fuse
CDFs that can be generalized to arbitrary scenes. A corresponding neural CDF
representation using multilayer perceptrons is also presented to obtain a
compact and continuous representation while improving computation efficiency.
We demonstrate the effectiveness of CDF with planar obstacle avoidance examples
and with a 7-axis Franka robot in inverse kinematics and manipulation planning
tasks.
