Motion planning, also path planning (also known as the navigation problem or the piano mover's problem) is a computational problem to find a sequence of valid configurations that moves the object from the source to destination. The term is used in computational geometry, computer animation, robotics and computer games. For example, consider navigating a mobile robot inside a building to a distant waypoint. It should execute this task while avoiding walls and not falling down stairs. A motion planning algorithm would take a description of these tasks as input, and produce the speed and turning commands sent to the robot's wheels. Motion planning algorithms might address robots with a larger number of joints (e.g., industrial manipulators), more complex tasks (e.g. manipulation of objects), different constraints (e.g., a car that can only drive forward), and uncertainty (e.g. imperfect models of the environment or robot). Motion planning has several robotics applications, such as autonomy, automation, and robot design in CAD software, as well as applications in other fields, such as animating digital characters, video game, architectural design, robotic surgery, and the study of biological molecules.
Concepts
A basic motion planning problem is to compute a continuous path that connects a start configuration S and a goal configuration G, while avoiding collision with known obstacles. The robot and obstacle geometry is described in a 2D or 3D workspace, while the motion is represented as a path in (possibly higher-dimensional) configuration space.
Configuration space A configuration describes the pose of the robot, and the configuration space C is the set of all possible configurations. For example:
If the robot is a single point (zero-sized) translating in a 2-dimensional plane (the workspace), C is a plane, and a configuration can be represented using two parameters (x, y). If the robot is a 2D shape that can translate and rotate, the workspace is still 2-dimensional. However, C is the special Euclidean group SE(2) = R2 × {\displaystyle \times } SO(2) (where SO(2) is the special orthogonal group of 2D rotations), and a configuration can be represented using 3 parameters (x, y, θ). If the robot is a solid 3D shape that can translate and rotate, the workspace is 3-dimensional, but C is the special Euclidean group SE(3) = R3 × {\displaystyle \times } SO(3), then the configuration requires 6 parameters: (x, y, z) for translation, and Euler angles (α, β, γ). If the robot is a fixed-base manipulator with N revolute joints (and no closed-loops), C is N-dimensional. If gimbal lock is not acceptable (eg. in RN where N ≥ 3 ) then the use of quaternions or other workarounds may be required, increasing the rotational dimensions or solution's complexity.
Free space The set of configurations that avoids collision with obstacles is called the free space Cfree. The complement of Cfree in C is called the obstacle or forbidden region. Often, it is prohibitively difficult to explicitly compute the shape of Cfree. However, testing whether a given configuration is in Cfree is efficient. First, forward kinematics determine the position of the robot's geometry, and collision detection tests if the robot's geometry collides with the environment's geometry.
Target space Target space is a subspace of free space which denotes where we want the robot to move to. In global motion planning, target space is observable by the robot's sensors. However, in local motion planning, the robot cannot observe the target space in some states. To solve this problem, the robot goes through several virtual target spaces, each of which is located within the observable area (around the robot). A virtual target space is called a sub-goal.
Obstacle space Obstacle space is a space that the robot can not move to. Obstacle space is not opposite of free space.
Algorithms Low-dimensional problems can be solved with grid-based algorithms that overlay a grid on top of configuration space, or geometric algorithms that compute the shape and connectivity of Cfree. Exact motion planning for high-dimensional systems under complex constraints is computationally intractable. Potential-field algorithms are efficient, but fall prey to local minima (an exception is the harmonic potential fields). Sampling-based algorithms avoid the problem of local minima, and solve many problems quite quickly. They are unable to determine that no path exists, but they have a probability of failure that decreases to zero as more time is spent. Sampling-based algorithms are currently considered state-of-the-art for motion planning in high-dimensional spaces, and have been applied to problems which have dozens or even hundreds of dimensions (robotic manipulators, biological molecules, animated digital characters, and legged robots).
Grid-based search Grid-based approaches overlay a grid on configuration space and assume each configuration is identified with a grid point. At each grid point, the robot is allowed to move to adjacent grid points as long as the line between them is completely contained within Cfree (this is tested with collision detection). This discretizes the set of actions, and search algorithms (like A*) are used to find a path from the start to the goal. These approaches require setting a grid resolution. Search is faster with coarser grids, but the algorithm will fail to find paths through narrow portions of Cfree. Furthermore, the number of points on the grid grows exponentially in the configuration space dimension, which make them inappropriate for high-dimensional problems. Traditional grid-based approaches produce paths whose heading changes are constrained to multiples of a given base angle, often resulting in suboptimal paths. Any-angle path planning approaches find shorter paths by propagating information along grid edges (to search fast) without constraining their paths to grid edges (to find short paths). Grid-based approaches often need to search repeatedly, for example, when the knowledge of the robot about the configuration space changes or the configuration space itself changes during path following. Incremental heuristic search algorithms replan fast by using experience with the previous similar path-planning problems to speed up their search for the current one.
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