12.3 Motion planning and configuration spaces

Motion planning is a crucial aspect of robotics, allowing machines to navigate complex environments. Configuration spaces provide a mathematical framework for representing robot positions and movements, simplifying the planning process by transforming physical obstacles into abstract constraints.

This topic connects to the broader chapter by showcasing how algebraic geometry applies to real-world robotics challenges. It demonstrates how mathematical concepts can be used to solve practical problems in robot motion, collision avoidance, and path planning.

Configuration space for motion planning

Concept and relevance

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• Configuration space (C-space) mathematically represents all possible configurations or poses of a robot, considering its degrees of freedom and motion constraints
• Each point in C-space represents a unique robot configuration, encompassing all achievable configurations
• C-space simplifies motion planning by reducing the problem to finding a path in a higher-dimensional space, solvable using various algorithms
• Obstacles in the robot's workspace are mapped into C-space, creating regions the robot must avoid to prevent collisions
• Motion planning in C-space involves finding a continuous collision-free path from the initial to the goal configuration

Representation and mapping

• Robot configurations are represented using generalized coordinates (joint angles for articulated robots, position and orientation for mobile robots)
• Each robot degree of freedom corresponds to a C-space dimension, resulting in high-dimensional spaces for complex robots
• Obstacles are mapped into C-space by considering the robot's geometry and the obstacle's shape and position
• The free space in C-space, where the robot can move without collisions, is defined as the complement of the obstacle regions

Representing robots and obstacles algebraically

Algebraic equations and inequalities

• Algebraic equations and inequalities define the boundaries and interior of obstacle regions in C-space
• Robot configurations can be represented using a set of generalized coordinates
• Algebraic representations allow for efficient collision checking and path planning algorithms
• The free space in C-space is defined as the complement of the obstacle regions
• Algebraic representations enable the application of mathematical tools and techniques for motion planning

Collision detection and avoidance

• Collision detection determines if a robot configuration intersects with any obstacle regions in C-space
• Algebraic collision detection algorithms evaluate equations and inequalities representing the robot and obstacles to check for intersections
• Common algebraic collision detection methods include the separating axis theorem, Gilbert-Johnson-Keerthi (GJK) algorithm, and linear programming approaches
• Collision avoidance algorithms generate collision-free paths in C-space by modifying the robot's trajectory or configuration
• Potential field methods create artificial potential functions in C-space (obstacles have repulsive potentials, goal has an attractive potential) to guide the robot

Collision detection and avoidance algorithms

Sampling-based methods

• Sampling-based methods randomly sample configurations in C-space and connect them to create collision-free paths
• Rapidly-exploring Random Trees (RRT) incrementally build a tree of configurations by randomly sampling points and extending the tree towards them
• Probabilistic Roadmaps (PRM) construct a graph of collision-free configurations and connect them using local planning techniques
• Sampling-based methods are probabilistically complete, meaning the probability of finding a solution approaches one as the number of samples increases
• These methods are effective in high-dimensional spaces and can handle complex robot geometries and obstacle shapes

Optimization-based approaches

• Optimization-based approaches formulate motion planning as an optimization problem, seeking to minimize a cost function while satisfying constraints
• Cost functions can include path length, energy consumption, smoothness, or other performance metrics
• Constraints include collision avoidance, joint limits, velocity and acceleration bounds, and task-specific requirements
• Convex optimization techniques (linear programming, quadratic programming) can efficiently solve motion planning problems with linear or quadratic cost functions and constraints
• Nonlinear optimization methods (sequential quadratic programming, interior point methods) handle more complex cost functions and constraints but may have higher computational complexity

Complexity and completeness of algebraic methods

Computational complexity

• Computational complexity refers to the time and space requirements of motion planning algorithms as problem size increases
• Complexity depends on C-space dimensionality, number and complexity of obstacle regions, and chosen algorithms
• Algebraic methods often have higher computational complexity compared to sampling-based approaches, especially in high-dimensional spaces
• The trade-off between completeness and computational complexity should be considered when selecting motion planning algorithms for specific applications
• Efficient implementations and approximations can help mitigate the computational burden of algebraic methods

Completeness guarantees

• Completeness is a property of motion planning algorithms that guarantees finding a solution if one exists or reporting that no solution exists
• Algebraic motion planning methods can be resolution complete, finding a solution if one exists at a given C-space resolution or discretization
• Resolution completeness depends on the granularity of the discretization and may require fine resolutions for narrow passages or tight constraints
• Probabilistic completeness is another notion, where the probability of finding a solution approaches one as the number of samples or iterations increases
• Sampling-based methods (RRT, PRM) are probabilistically complete, while some algebraic methods may not provide completeness guarantees
• Completeness is important for mission-critical applications where finding a solution is crucial, but may be relaxed for time-sensitive or approximate planning scenarios