# Sampling-based methods for motion planning with constraints

** Published:**

This paper considers geometrical constraints that reduce the number of degrees of freedom.

Such a constraint defines a lower dimensional manifold in the configuration space. \(X = \{q \in Q|F(q)=0\}\) $F(q)$ is the constraint function. An example of such constraint is keeping the EE at some orientation during the motion.

Since the constraint manifold is a lower dimensional space embedded in the C-space, the relative measure (volume) is zero compared to C-space. Therefore, randomly sampled configuration will not lie on the constraint manifold. Basically, there are mainly five approaches to handle the constraints:

- Relaxation: set a tolerance which increases the volume of constraint space. It’s like adding additional dimensions to the constraint manifold.
- Projection: Use the Jacobian of $F(q)$ to reduce the error (distance) to the constraint manifold.
- Tangent space: Locally linearize the manifold at the current configuration.
- Atlas: Use a set of linearized planes to approximate the manifold?
- Reparameterization: Only applicable to some special cases.

To be continued…

### Reference

Kingston, Zachary, Mark Moll, and Lydia E. Kavraki. “Sampling-based methods for motion planning with constraints.” Annual review of control, robotics, and autonomous systems 1 (2018): 159-185.