📐Computational Geometry Unit 7 – Motion Planning & Collision Detection

Key Concepts and Definitions

• Motion planning involves finding a valid path for a robot or object from a start configuration to a goal configuration while avoiding obstacles
• Configuration space (C-space) represents all possible states or configurations of a robot, where each point corresponds to a unique pose or position and orientation
• Degrees of freedom (DOF) refer to the number of independent parameters needed to fully specify the configuration of a robot or object
• Obstacle region in C-space corresponds to configurations where the robot collides with obstacles in the workspace
• Free space consists of all configurations where the robot does not collide with any obstacles
  • Feasible path lies entirely within the free space and connects the start and goal configurations
• Holonomic constraints are constraints that can be expressed as equations involving only the configuration variables and time, such as joint limits or velocity bounds
• Non-holonomic constraints cannot be expressed using only configuration variables and time, such as rolling without slipping for wheeled robots

Fundamental Algorithms

• Sampling-based algorithms rely on randomly sampling the configuration space to build a graph or tree of feasible configurations and paths
  • Rapidly-exploring Random Trees (RRT) incrementally grow a tree from the start configuration by randomly sampling points and extending the tree towards them
  • Probabilistic Roadmaps (PRM) construct a graph by randomly sampling configurations and connecting nearby samples with local planners
• Combinatorial planning algorithms discretize the configuration space into cells and search for a path using graph search techniques
  • Cell decomposition methods partition the free space into simple regions (cells) and build a connectivity graph representing adjacency between cells
  • Visibility graphs connect vertices of obstacle polygons and the start/goal configurations, ensuring edges do not intersect obstacles
• Potential field methods define an artificial potential function over the configuration space, with the goal having an attractive potential and obstacles having repulsive potentials
  • Gradient descent is used to guide the robot towards the goal while avoiding obstacles
• Optimization-based approaches formulate motion planning as an optimization problem, minimizing a cost function subject to constraints
  • Trajectory optimization techniques (CHOMP, TrajOpt) iteratively refine an initial trajectory to minimize a cost function while satisfying constraints

Motion Planning Techniques

• Kinodynamic planning considers the robot's dynamics and differential constraints, such as velocity and acceleration limits, in addition to geometric constraints
• Nonholonomic planning deals with systems subject to non-integrable velocity constraints, such as wheeled robots or underactuated systems
• Manipulation planning involves planning for robotic arms to grasp, manipulate, and transport objects while avoiding collisions
  • Multi-modal planning combines different modes of operation (transit, transfer, push, pull) to find a sequence of actions to achieve the task
• Multi-robot planning coordinates the motions of multiple robots to avoid collisions and achieve a common goal
  • Centralized approaches plan for all robots simultaneously, considering their interactions and constraints
  • Decentralized approaches have each robot plan independently while communicating and negotiating to avoid conflicts
• Adaptive and learning-based planning methods use machine learning techniques to improve planning performance over time or adapt to changing environments
  • Reinforcement learning allows robots to learn optimal policies through trial-and-error interactions with the environment
• Uncertainty-aware planning considers uncertainties in robot motion, sensing, and environment models to generate robust plans
  • Belief space planning maintains a probability distribution over the robot's state and plans in the space of beliefs
• Integrated task and motion planning combines high-level task planning with low-level motion planning to solve complex, multi-step problems

Collision Detection Methods

• Geometric collision detection checks for intersections between geometric representations of the robot and obstacles
  • Bounding volume hierarchies (BVH) recursively partition objects into simpler bounding volumes (spheres, boxes, capsules) to accelerate collision queries
  • Signed distance fields (SDF) store the signed distance to the nearest obstacle surface at each point in space, allowing fast collision and proximity queries
• Continuous collision detection (CCD) checks for collisions along a continuous path or trajectory, considering the swept volume of the robot's motion
  • Conservative advancement incrementally steps along the path, checking for collisions at each step and adjusting the step size based on the distance to the nearest obstacle
• Narrow-phase collision detection performs detailed intersection tests between pairs of geometry primitives (triangles, polygons) to determine exact contact points and normals
  • Gilbert-Johnson-Keerthi (GJK) algorithm computes the minimum distance between convex shapes by iteratively refining a simplex in the Minkowski difference of the shapes
  • Separating Axis Theorem (SAT) checks for a separating axis between two convex shapes by projecting them onto a set of candidate axes and checking for overlap
• Broad-phase collision detection quickly identifies potentially colliding pairs of objects using spatial partitioning or bounding volume tests, reducing the number of pairs requiring narrow-phase checks
  • Spatial hashing divides the space into a grid of cells and assigns objects to the cells they overlap, allowing fast retrieval of nearby objects
  • Sweep and prune maintains sorted lists of object bounding boxes along each axis and checks for overlaps between pairs of objects with overlapping projections

Data Structures and Representations

• Occupancy grids discretize the environment into a grid of cells, where each cell stores information about the presence of obstacles or the probability of occupancy
• Octrees recursively partition the space into eight octants, allowing adaptive resolution and efficient spatial queries
  • Each node represents a cubic volume and has eight children corresponding to the eight octants
  • Nodes are subdivided until a desired resolution is reached or they contain only free space or obstacles
• Voxel grids are 3D extensions of occupancy grids, representing the environment as a regular grid of cubic cells (voxels)
• Point clouds are sets of 3D points, often obtained from depth sensors or laser scanners, representing the surface geometry of the environment
  • Organized point clouds have a regular 2D grid structure, while unorganized point clouds have no specific order
• Meshes represent surfaces as a collection of connected polygons (usually triangles), providing a more detailed and accurate representation compared to voxels or point clouds
  • Half-edge data structure efficiently represents the connectivity and adjacency information between vertices, edges, and faces of a mesh
• Implicit surfaces define the surface as the zero level-set of a function, allowing smooth and continuous representations
  • Signed distance functions (SDF) store the signed distance to the surface at each point, with negative values inside and positive values outside the surface

Practical Applications

• Autonomous navigation for mobile robots, drones, and self-driving cars
  • Planning safe and efficient paths in complex, dynamic environments while following traffic rules and avoiding obstacles
• Robotic manipulation in manufacturing, assembly, and pick-and-place tasks
  • Planning motions for robotic arms to grasp, manipulate, and transport objects while avoiding collisions with the environment and other robots
• Surgical robotics and medical procedures
  • Planning safe and precise trajectories for surgical instruments to reach target locations while avoiding sensitive anatomical structures
• Computer animation and virtual simulations
  • Generating realistic and collision-free motions for characters and objects in virtual environments, such as video games or training simulations
• Protein folding and molecular docking in computational biology
  • Predicting the 3D structure of proteins by finding low-energy conformations and analyzing the feasibility of protein-ligand binding poses
• Aerospace and spacecraft motion planning
  • Planning trajectories for satellites, spacecraft, and robotic manipulators in microgravity environments, considering fuel efficiency and collision avoidance
• Crowd simulation and pedestrian modeling
  • Simulating the motion of large crowds or pedestrian flows in urban environments, public spaces, or evacuation scenarios, considering social interactions and collision avoidance

Advanced Topics and Current Research

• Topological motion planning considers the topological properties of the configuration space, such as homotopy classes of paths, to find diverse and globally optimal solutions
• Integrated perception and planning incorporates real-time sensor data and updates the environment representation and motion plans on-the-fly
  • Active perception strategies plan sensor movements to actively gather information and reduce uncertainty
• Learning-based collision detection uses machine learning techniques, such as neural networks, to learn implicit representations of the environment and quickly predict collisions
• Differentiable motion planning allows end-to-end learning of motion planners by making the planning algorithms differentiable and enabling gradient-based optimization
• Cooperative and multi-agent planning extends motion planning techniques to systems with multiple interacting agents, considering communication, coordination, and game-theoretic aspects
• Semantic motion planning incorporates high-level semantic information, such as object affordances and task-specific constraints, to generate more efficient and interpretable plans
• Real-time and anytime planning algorithms provide fast, suboptimal solutions that improve over time, allowing robots to respond quickly to dynamic environments
• Uncertainty quantification and robust planning develop techniques to quantify and propagate uncertainties in motion planning and generate plans that are robust to disturbances and modeling errors

Challenges and Future Directions

• Scalability to high-dimensional and complex configuration spaces
  • Developing efficient sampling strategies, heuristics, and dimensionality reduction techniques to handle the curse of dimensionality
• Real-time performance for dynamic and unpredictable environments
  • Designing fast, incremental, and anytime planning algorithms that can adapt to changing conditions and provide safety guarantees
• Integration with perception, control, and decision-making modules
  • Developing unified frameworks that seamlessly integrate motion planning with other aspects of robot autonomy, considering uncertainties and feedback loops
• Generalization and transfer learning across tasks and environments
  • Learning reusable and transferable representations and skills that can be adapted to new situations with minimal fine-tuning or re-training
• Interpretability and explainability of planned motions
  • Developing methods to explain and visualize the reasoning behind planned motions, building trust and facilitating human-robot interaction
• Ethical and safety considerations in autonomous systems
  • Incorporating ethical principles and safety constraints into motion planning algorithms, ensuring compliance with regulations and societal norms
• Benchmarking and standardization of evaluation metrics
  • Establishing common benchmarks, datasets, and performance metrics to facilitate comparisons and reproducibility of motion planning algorithms
• Collaboration between academia and industry
  • Fostering knowledge exchange and technology transfer between research institutions and companies to accelerate the deployment of motion planning techniques in real-world applications