Sampling-Based Planning
Sampling-based planning is a family of methods for finding a path through a space that is far too vast to examine point by point. The space in question is the robot's set of all possible poses — every combination of joint angles or positions it could strike — which for an arm with several joints is a huge, many-dimensional region impossible to map out square by square. Instead of carving that whole region into a fine grid, a sampling-based planner throws down random poses like darts on a board, keeps the ones where the robot would not crash into anything, and gradually stitches these scattered safe poses into a network it can travel along.
The reason this works so well is that it sidesteps the curse of dimensionality — the explosion in size that happens when you try to grid up a space with many dimensions. A grid that is fine enough to be useful would need astronomically many squares once you have six or more joints, but a handful of well-placed random samples can capture the shape of the free space without ever enumerating it. Each new sample is checked for collisions and, if safe, connected to nearby samples by short links that are themselves checked to be collision-free, so the network only ever contains motions the robot can actually make.
The trade-off is that randomness gives up some guarantees. A sampling-based planner is usually probabilistically complete, meaning that if a path exists it will eventually be found given enough samples — but 'eventually' is not 'immediately', and on any single run it may return no path even when one is there, or return a path that is valid but not the shortest. The two best-known members of this family are the probabilistic roadmap, which builds a reusable map of the whole space, and the rapidly-exploring random tree, which grows a single tree outward from the start toward the goal.
To plan how a six-joint arm reaches behind a clutter of boxes, a planner samples thousands of random arm poses, discards those that would clip a box, and links the survivors. A clear chain of safe links from the start pose to the grasp pose becomes the motion the arm executes — found without ever gridding the arm's vast pose space.
Random poses plus collision checks build a safe network through a space too big to grid.
These methods plan in configuration space (the space of the robot's poses), not in the ordinary 3-D space around it, which is what lets them handle arms with many joints.