The provided texts constitute a collection of research papers concerning various facets of algorithmic foundations in robotics. Several papers explore motion planning for single and multiple robots in complex environments, addressing challenges like optimality, collision avoidance, handling dynamic obstacles, and incorporating human guidance. Other works investigate localization and mapping techniques for robot swarms and individual agents under uncertainty, often utilizing probabilistic methods. Furthermore, the collection covers advanced topics such as task and motion planning integration, manipulation in contact, the theoretical underpinnings of robot control, and the application of topological concepts to robotic problems like coverage and knot manipulation. Finally, some papers introduce novel algorithms and provide theoretical analyses of their completeness, optimality, and efficiency in addressing specific robotics challenges.

Algorithmic Foundations of Robotics XI: Study Guide

Quiz

1. What is the primary challenge addressed in “Efficient Multi-robot Motion Planning for Unlabeled Discs in Simple Polygons”? Briefly describe the approach taken to tackle this challenge.
2. In “Navigation of Distinct Euclidean Particles via Hierarchical Clustering,” what is the significance of hierarchical clustering in the context of multi-agent navigation? Explain the concept of an “admissible cluster.”
3. According to “Coalition Formation Games for Dynamic Multirobot Tasks,” why are coalition formation games relevant for coordinating multiple robots in dynamic environments? Provide a brief example of a scenario where this approach would be beneficial.
4. What is the core idea behind “Computing Large Convex Regions of Obstacle-Free Space Through Semidefinite Programming”? How does semidefinite programming help in achieving this?
5. In “A Region-Based Strategy for Collaborative Roadmap Construction,” how does the approach leverage regions to facilitate the construction of a roadmap for robot motion planning? What are the advantages of this collaborative strategy?
6. According to “Efficient Sampling-Based Approaches to Optimal Path Planning in Complex Cost Spaces,” what are the key challenges when planning optimal paths in such spaces? Briefly describe a sampling-based technique used to address these challenges.
7. What is the main focus of “Real-Time Predictive Modeling and Robust Avoidance of Pedestrians with Uncertain, Changing Intentions”? Briefly explain how predictive modeling contributes to robust avoidance.
8. In “FFRob: An Efficient Heuristic for Task and Motion Planning,” what is the central goal of the proposed heuristic? How does it aim to improve the efficiency of task and motion planning?
9. According to “Fast Nearest Neighbor Search in SE(3) for Sampling-Based Motion Planning,” why is nearest neighbor search in SE(3) a critical operation in sampling-based motion planning? What makes it challenging, and what is a potential approach to improve its speed?
10. What is the problem of “Trackability with Imprecise Localization” concerned with? Briefly describe a scenario where a robot might face challenges related to trackability due to imprecise localization.

Quiz Answer Key

1. The paper addresses the challenge of efficiently planning collision-free motions for multiple identical (unlabeled) disc-shaped robots within a simple polygon. Their approach involves decomposing the free space and constructing a graph that captures the connectivity of feasible configurations, allowing for efficient path finding.
2. Hierarchical clustering is used to group the particles and simplify the control strategy by defining collective behaviors based on cluster properties. An “admissible cluster” for a given configuration signifies a cluster where the particles within it exhibit a certain level of consensus, quantified by the non-positive value of $\eta_{i,I,\tau}(x)$.
3. Coalition formation games are relevant because they provide a framework for robots to autonomously decide which tasks to undertake and with which other robots to collaborate, especially when tasks and robot capabilities change over time. For example, in a search and rescue scenario, robots might form coalitions to cover a larger area or to combine specialized sensing capabilities.
4. The core idea is to represent the obstacle-free space as a union of large convex regions by formulating constraints on the regions using semidefinite programming (SDP). SDP allows for optimization over convex sets of matrices, enabling the computation of maximal volume ellipsoids or other convex shapes that are guaranteed to be collision-free.
5. The region-based strategy aims to improve collaborative roadmap construction by having robots independently explore local regions of the environment and then collaboratively merge these local roadmaps based on the connectivity of the regions. This can lead to more efficient exploration and a more robust global roadmap compared to purely centralized or decentralized approaches.
6. Planning optimal paths in complex cost spaces is challenging due to the high dimensionality of the configuration space, the presence of obstacles or regions with high costs, and the difficulty in efficiently exploring the space to find low-cost paths. Sampling-based techniques like RRT* address this by randomly sampling the configuration space and iteratively connecting these samples to build a graph that converges to an optimal path as the number of samples increases.
7. The paper focuses on enabling robots to navigate safely in the presence of pedestrians by predicting their future motion and intentions, which are often uncertain and changing. Predictive modeling helps in anticipating potential collisions and allows the robot to plan robust avoidance maneuvers that take into account the uncertainty in pedestrian behavior.
8. The central goal of FFRob is to provide an efficient heuristic for solving combined task and motion planning problems, which are generally computationally expensive. The heuristic likely aims to decompose the problem or use abstraction to reduce the search space, allowing for faster solutions compared to traditional integrated approaches.
9. Nearest neighbor search in SE(3) (the space of 3D rigid body poses) is crucial for many sampling-based motion planning algorithms, as it’s used to find the closest existing states to newly sampled states for connection and extension. It’s challenging due to the non-Euclidean nature of SE(3) and the need for metrics that consider both position and orientation. Techniques like specialized data structures (e.g., k-d trees adapted for SE(3)) and efficient distance metrics are used to improve speed.
10. The problem of “Trackability with Imprecise Localization” deals with ensuring that a robot can reliably track a desired trajectory or goal even when its own localization (knowledge of its pose) is uncertain or noisy. A robot navigating in a GPS-denied environment or relying on noisy sensor data might face challenges in accurately following a planned path or reaching a target location due to imprecise localization.

Essay Format Questions

1. Compare and contrast sampling-based and optimization-based approaches to motion planning as represented in the provided excerpts. Discuss the strengths and weaknesses of each approach in the context of different robotic tasks and environments. Refer to specific papers to support your arguments.
2. Several papers in the collection address multi-robot systems. Analyze the different coordination strategies presented, such as coalition formation, hierarchical clustering, and collaborative roadmap construction. Discuss the conditions under which each strategy is most appropriate and the challenges associated with their implementation.
3. Uncertainty plays a significant role in many robotic applications. Discuss how different forms of uncertainty (e.g., in sensor measurements, environment models, or agent intentions) are addressed in the featured research. Provide examples from at least three different papers.
4. The concept of “optimality” appears in several paper titles. Critically evaluate what constitutes an “optimal” solution in the context of robot motion planning and control, considering factors such as path length, time, energy consumption, and robustness. Refer to specific papers that define and pursue optimality in different ways.
5. Discuss the challenges and advancements in addressing the complexity of robot motion planning in high-dimensional configuration spaces, as evidenced by the variety of topics covered in the excerpts. Consider the role of sampling, abstraction, heuristics, and different representations of the state space in managing this complexity.

Glossary of Key Terms

• Configuration Space (C-space): The space that represents all possible poses (position and orientation) of a robot or a system. Each point in C-space corresponds to a unique configuration of the robot.
• Free Space (Cfree): The subset of the configuration space that corresponds to configurations where the robot is not in collision with any obstacles in the environment.
• Motion Planning: The problem of finding a valid (collision-free) path for a robot to move from a start configuration to a goal configuration in its environment.
• Sampling-Based Motion Planning: A class of motion planning algorithms that explore the configuration space by randomly sampling points and connecting them to build a roadmap or a tree, which is then searched for a path. Examples include RRT and PRM.
• Optimal Path Planning: Motion planning with the objective of finding a path that not only is collision-free but also minimizes a certain cost function, such as path length, travel time, or energy consumption.
• Multi-robot Motion Planning: The problem of coordinating the motion of multiple robots to achieve individual or collective goals while avoiding collisions among themselves and with the environment.
• Collision Detection: The process of determining whether a robot in a given configuration intersects with any obstacles or other robots in the environment.
• Degrees of Freedom (DOF): The number of independent parameters that define the configuration of a robot or a system.
• Kinematics: The study of motion without regard to the forces causing it. In robotics, it often refers to the relationship between the robot’s joint angles and the position and orientation of its end-effector or other parts.
• Dynamics: The study of motion in relation to the forces and torques that cause it. In robotics, it involves modeling the robot’s equations of motion, taking into account factors like inertia, friction, and gravity.
• Heuristic: A problem-solving approach that uses practical methods or shortcuts to produce solutions that may not be optimal but are sufficient for a given set of constraints.
• Semidefinite Programming (SDP): A type of convex optimization problem involving the optimization of a linear objective function over the intersection of the cone of positive semidefinite matrices with an affine space.
• Roadmap: A graph representing the connectivity of the free space, where nodes correspond to collision-free configurations and edges represent feasible paths between them.
• Nearest Neighbor Search: An algorithmic problem of finding the point in a set that is closest (according to some distance metric) to a given query point.
• SE(3): The special Euclidean group in 3D, representing the space of rigid body motions (translations and rotations) in three-dimensional space.
• Localization: The problem of determining a robot’s pose (position and orientation) within its environment.
• Control Policy: A rule or a function that determines the actions (control inputs) a robot should take based on its current state and/or the state of the environment.
• Stochastic Dynamics: A model of how a system’s state evolves over time that includes random elements or noise.
• Temporal Logic (LTL): A type of modal logic used to describe and reason about sequences of events in time. It is often used to specify complex mission requirements for robots.
• Bayesian Approach: A statistical method that uses Bayes’ theorem to update the probability for a hypothesis as more evidence or information becomes available.
• Gaussian Process (GP): A probabilistic kernel-based model that defines a distribution over functions. It is often used for regression and classification tasks, especially when dealing with uncertainty.
• Dynamic Programming: An optimization method that breaks down a complex problem into smaller overlapping subproblems, solves each subproblem only once, and stores the solutions to avoid redundant computations.
• Feedback Control: A control strategy where the control actions are based on the difference between the desired state and the actual state of the system.
• Lyapunov Function: A scalar function used to analyze the stability of a dynamical system. Its properties (e.g., being positive definite and having a negative semi-definite derivative along the system’s trajectories) can guarantee stability.

Briefing Document: Algorithmic Foundations of Robotics XI

This briefing document summarizes the main themes and important ideas presented in the table of contents and selected excerpts from “Algorithmic Foundations of Robotics XI.” The collection of papers covers a wide range of topics within robotics, broadly focusing on motion planning, control, perception, and manipulation in both single and multi-robot systems.

Main Themes:

Several overarching themes emerge from the listed papers:

• Efficient and Optimal Motion Planning: A significant portion of the research focuses on developing algorithms for finding efficient, and ideally optimal, paths and trajectories for robots in complex environments. This includes addressing challenges such as high-dimensional state spaces, kinodynamic constraints, temporal goals, and dynamic obstacles.
• Multi-Robot Systems: Many papers explore coordination, planning, and control in systems with multiple robots. Topics range from efficient motion planning for unlabeled discs and coalition formation to cooperative roadmap construction and optimal task allocation in delivery systems.
• Handling Uncertainty and Stochasticity: Several contributions address the inherent uncertainty in robotic systems and environments. This includes predictive modeling of pedestrian intentions, motion planning under uncertainty, active information gathering for localization, and planning in stochastic environments.
• Advanced Algorithmic Techniques: The papers leverage a diverse set of advanced algorithmic techniques, including sampling-based methods (RRT, PRM), optimization (semidefinite programming, quadratic programming, trajectory optimization), hierarchical clustering, graph search algorithms, and formal methods (LTL, automata).
• Real-Time and Reactive Planning: Several works emphasize the need for robots to operate in dynamic environments and respond to changes in real-time. This includes real-time motion planning with unpredictable obstacles and robust avoidance strategies.
• Manipulation and Interaction with Objects: Some papers delve into the complexities of robot manipulation, including orienting parts with shape variation, quasi-static whole-body manipulation, and even knot manipulation.

Important Ideas and Facts from Excerpts:

Here are some key ideas and facts highlighted in the provided excerpts, with direct quotes where relevant:

1. Efficient Multi-robot Motion Planning for Unlabeled Discs:

• This paper tackles the problem of planning motion for multiple indistinguishable disc-shaped robots in simple polygonal environments.
• Lemma 9 states: “The combinatorial complexity of D ⋃x∈S∪T D∗(x), is O(m + n).” This suggests an efficient approach to characterizing the free space by considering the union of certain disc-based regions related to start and target configurations.
• The paper discusses constructing a graph Gi by selecting representative points βi(x) on the boundary of collision discs and connecting start and target positions. This hints at a graph-based approach to solving the multi-robot motion planning problem.

2. Navigation of Distinct Euclidean Particles via Hierarchical Clustering:

• This work proposes using hierarchical clustering to navigate a set of distinct particles.
• It introduces the concept of “hierarchy-invariant vector fields,” defined as: “FHC(τ ) : = { f : Conf ( R d , J ) → ( R d )J ∣∣∣ϕt ( S (τ ) ) ⊂ S̊ (τ ) , t > 0 } , (4)” These vector fields ensure that certain clustered configurations remain within their “stratum” under the induced flow.
• The paper defines “admissible (valid)” clusters based on the inequality: “ηi,I,τ (x) : = ( xi − mI,τ (x) )TsI,τ (x) ≤ 0 for all i ∈ I . (8)” This condition likely plays a crucial role in the control strategy based on hierarchical structure.
• The “consensus ball” BQ(x) is introduced as “the largest open ball…centered at c (x|Q) so that for any y ∈ YQ ( x, BQ (x) ) and γ ∈ {σ, τ } every cluster D ∈ {Q,Pr (Q, γ)} \ {P} of γ are partially admissible for y|Q.” This defines a region around a partial configuration where certain admissibility conditions are maintained.
• “Portal Maps” are defined as a continuous map between different hierarchical structures, aiming to connect different organizational levels of the particle system.

3. Active Control Strategies for Discovering and Localizing Devices:

• This paper focuses on actively controlling a robot team to discover and localize devices with uncertain locations.
• It uses “mutual information” as a metric to quantify the information gained about the device locations through measurements: “MI[x, zτ (cτ )] = D∑d=1 MI [ xd ; zd τ ] = D∑d=1 H [ zd τ ] − H [ zd τ | xd ] (2)” This highlights the information-theoretic approach to active perception.
• A similar concept of mutual information is applied to discrete grid cells to localize devices within a grid: “MI[g, qτ ] = G∑i=1 MI [ gi , qi τ ] = G∑i=1 H [ qi τ ] − H [ qi τ | gi ] (4)” This demonstrates the adaptability of the mutual information metric to different representations of uncertainty.

4. Localization without Coordination:

• This paper presents a distributed algorithm for robot localization that does not require explicit coordination.
• Algorithm 1 outlines the steps involved, including broadcasting odometry information and then “find θ̂wk |uk , φ̂wk |uk such that (2) holds ∀ j ∈ I“. Equation (2) likely represents a constraint equation based on relative measurements and odometry, allowing each robot to estimate the pose of its neighbors.

5. Computing Large Convex Regions of Obstacle-Free Space Through Semidefinite Programming:

• This paper uses semidefinite programming to find large convex regions that are free of obstacles.
• The method involves finding an ellipsoid E and then iteratively finding a separating hyperplane between the ellipsoid and each obstacle: “We label the point of intersection between Eα∗ and j as x∗. We can then compute a hyperplane, a∗ j x = b, with a j ∈ IRn and b j ∈ IR which is tangent to Eα∗ and which passes through x∗.” This process refines the convex free space representation.

6. Real-Time Predictive Modeling and Robust Avoidance of Pedestrians with Uncertain, Changing Intentions:

• This work deals with predicting pedestrian behavior and enabling robots to avoid them robustly, considering the uncertainty in their intentions.
• The paper uses a probabilistic approach to model motion patterns and assign trajectories to existing or new patterns based on their likelihood: “p(zi = j |t i ,α, θGP x, j , θGP y, j ) ∝ p(t i |b j ) ( n j N − 1 + α ) , (4)” This Bayesian framework allows for adapting to changing pedestrian behavior.

7. FFRob: An Efficient Heuristic for Task and Motion Planning:

• This paper introduces an efficient heuristic for integrated task and motion planning.
• It defines the concept of an “operator” with preconditions, effects (add and delete lists of literals), and a function that maps detailed states: “successor(s, a) ≡ 〈s.L ∪ a.epos \ a.eneg, a. f (s)〉 .” This is a standard representation in planning systems.

8. Fast Nearest Neighbor Search in SE(3) for Sampling-Based Motion Planning:

• This paper addresses the challenge of efficient nearest neighbor search in the six-dimensional space SE(3) (rigid body poses), which is crucial for sampling-based motion planning algorithms.
• It defines a distance metric “DIST Rm P3(q1,q2) = αDISTRm (q1,q2) + DISTP3(q1,q2).” which combines translational and rotational distances with a weighting factor α.
• The paper introduces a “DynamicKDSearch” algorithm (Algorithm 3) that seems to adaptively refine the search structure based on the query point and the distribution of configurations.
• The paper discusses splitting criteria for the search structure, including splitting at the midpoint or at a hyperplane intersecting the point being inserted.

9. Trackability with Imprecise Localization:

• This paper likely investigates the conditions under which a robot can track a target despite having imprecise localization capabilities.
• Figure 6 illustrates a “gadget construction” related to intersections and path lengths, suggesting an analysis of how localization uncertainty affects the ability to follow or remain within a certain distance of a trajectory.

10. Kinodynamic RRTs with Fixed Time Step and Best-Input Extension Are Not Probabilistically Complete:

• This paper presents a theoretical result showing that a specific variant of the RRT (Rapidly-exploring Random Tree) algorithm, when used with a fixed time step and a best-input extension strategy for systems with kinodynamic constraints, does not guarantee probabilistic completeness (the ability to find a solution path with probability approaching one as the number of samples increases).
• The problem formulation defines the system dynamics: “ẋ = f (x, u) (1)” and the goal is to find a control trajectory that satisfies these constraints, avoids collisions, and reaches the goal set.

11. Collision Prediction Among Rigid and Articulated Obstacles with Unknown Motion:

• This paper addresses the challenging problem of predicting collisions with moving obstacles whose motion is unknown.

12. Asymptotically Optimal Stochastic Motion Planning with Temporal Goals:

• This work focuses on motion planning for stochastic systems with goals specified using temporal logic (LTL).
• It defines the semantics of co-safe LTL formulas over infinite traces: “Let σ = {τi }∞i=0 denote an infinite trace… The notation σ |= φ denotes that the trace σ satisfies co-safe formula φ…” This provides a formal way to specify complex mission requirements that involve sequences of states or events.
• The problem is framed as finding a policy that satisfies the temporal goal while minimizing risk or cost in a stochastic environment.

13. Resolution-Exact Algorithms for Link Robots:

• This paper likely discusses motion planning algorithms for robots composed of links, aiming for solutions that are exact with respect to the discretization resolution.

14. Optimal Trajectories for Planar Rigid Bodies with Switching Costs:

• This paper investigates finding optimal trajectories for planar rigid bodies where there are costs associated with switching between different modes of motion or control inputs.

15. Maximum-Reward Motion in a Stochastic Environment: The Nonequilibrium Statistical Mechanics Perspective:

• This paper approaches the problem of motion planning for maximum reward in a stochastic environment using concepts from nonequilibrium statistical mechanics.
• Equation (2) relates the probability of finding a near-optimal path to the expected reward and a concentration term, suggesting a probabilistic analysis of performance.

16. Optimal Path Planning in Cooperative Heterogeneous Multi-robot Delivery Systems:

• This paper deals with finding optimal paths for a team of diverse robots (heterogeneous) cooperating to perform delivery tasks.
• The problem is modeled using a graph with different types of edges representing street and flight movements: “The edge set, E , is a unionof twomutually exclusive subsets, E = Ew∪Ed . The set Ew contains directed street edges… The set Ed contains pairs of bidirectional flight edges…” This graph-based formulation allows for capturing the different capabilities of the robots.
• The paper mentions a transformation from the Traveling Salesperson Problem (TSP) to their “Heterogeneous Delivery Problem (HDP),” suggesting a connection to classical combinatorial optimization problems.

17. Composing Dynamical Systems to Realize Dynamic Robotic Dancing:

• This work explores how to combine different dynamical systems to create complex and coordinated motions, specifically for robotic dancing.
• Equation (6) defines the desired and actual outputs for the “Single Support” phase of a bipedal robot, relating the robot’s configuration to desired foot placements and joint angles.

18. The Lion and Man Game on Convex Terrains:

• This paper likely analyzes a pursuit-evasion game (“Lion and Man”) played on convex terrains, focusing on strategies and conditions for capture.

19. RRTX: Real-Time Motion Planning/Replanning for Environments with Unpredictable Obstacles:

• This paper presents RRTX, an extension of the RRT algorithm designed for real-time replanning in environments where obstacles may appear or move unpredictably.
• Algorithms 2, 3, and 4 describe procedures for adding vertices, culling neighbors, and rewiring the search tree, highlighting the dynamic and reactive nature of the algorithm.
• Proposition 3 suggests that as the number of nodes increases, the distance between a new node and its parent in the RRTX tree tends to zero.

20. Orienting Parts with Shape Variation:

• This paper addresses the problem of manipulating parts with slight variations in their shape to achieve a desired orientation.
• Definition 2 classifies “p-stable angles” into R-type and L-type based on the behavior of a radius function, which likely characterizes the stability of orientations.
• Algorithms 1 and 2 outline procedures for constructing critical instances and computing the smallest possible orientation set, suggesting a geometric and analytical approach to solving the part orientation problem.

21. Smooth and Dynamically Stable Navigation of Multiple Human-Like Robots:

• This work focuses on enabling multiple humanoid robots to navigate smoothly and maintain dynamic stability.
• Equation (2) defines the “AVOδ,τ AB” (Avoidance Velocity Obstacle) between two robots, representing the set of relative velocities that would lead to a collision within a time horizon τ, considering acceleration control parameters.

22. Scaling up Gaussian Belief Space Planning Through Covariance-Free Trajectory Optimization and Automatic Differentiation:

• This paper tackles the challenge of planning in belief space (the space of probability distributions over the robot’s state) for systems with Gaussian uncertainty.
• Equations (3a-d) describe the Kalman filter update equations used to propagate the robot’s belief state over time.
• The paper also presents dynamic models for a two-link manipulator (Eq. 7) and a unicycle robot with sensor noise (Eq. 8), demonstrating the application of their belief space planning approach to different robotic systems.

23. Planning Curvature and Torsion Constrained Ribbons in 3D with Application to Intracavitary Brachytherapy:

• This paper focuses on planning paths for flexible instruments, modeled as curvature and torsion constrained ribbons in 3D space, with a specific application in medical brachytherapy.
• Equation (3) relates the derivatives of the ribbon’s Frenet-Serret frame (tangent, normal, binormal) to its curvature κt, torsion τt, and linear velocity vt.

24. A Quadratic Programming Approach to Quasi-Static Whole-Body Manipulation:

• This paper uses quadratic programming to solve problems of quasi-static manipulation involving the whole body of the robot.
• Equation (1) relates the velocity of the world frame center of mass to the robot’s base velocity and the joint velocities of its manipulators.
• Equation (4) defines the “base Jacobian,” and Equation (5) relates the center of mass velocity to the joint velocities via the “center of mass Jacobian.”

25. On-line Coverage of Planar Environments by a Battery Powered Autonomous Mobile Robot:

• This paper addresses the problem of autonomously covering a planar environment with a mobile robot that has limited battery power.

26. Finding a Needle in an Exponential Haystack: Discrete RRT for Exploration of Implicit Roadmaps in Multi-robot Motion Planning:

• This work presents a discrete version of the RRT algorithm for exploring implicit roadmaps in the context of multi-robot motion planning, potentially addressing the combinatorial complexity of such problems.

27. Stochastic Extended LQR: Optimization-Based Motion Planning Under Uncertainty:

• This paper introduces a stochastic extension of the Linear-Quadratic Regulator (LQR) framework for optimization-based motion planning under uncertainty.
• Equations (3) describe the inverse discrete dynamics of the system, and Equation (4) defines the cost function, which includes both path costs and a final cost.
• The paper outlines an iterative forward and backward value iteration process (lines 6-10 and 13-19 in Algorithm 1) to solve the stochastic optimal control problem.

28. An Approximation Algorithm for Time Optimal Multi-Robot Routing:

• This paper develops an approximation algorithm for finding time-optimal routes for multiple robots.

29. Decidability of Robot Manipulation Planning: Three Disks in the Plane:

• This paper investigates the theoretical decidability of motion planning for manipulating three disc-shaped robots in a planar environment.
• The concept of a “stratified configuration space” is introduced, where the space is decomposed into regular submanifolds based on constraints: “Si1i2…im = Φ−1 i1 (0) ∩ Φ−1 i2 (0) ∩ . . . Φ−1 im (0)“.
• The paper refers to “stratified controllability” as a condition for the system to be able to move in any direction within the configuration space.

30. A Topological Perspective on Cycling Robots for Full Tree Coverage:

• This paper takes a topological approach to analyze the problem of using cycling robots to achieve complete coverage of a tree-structured environment.
• Figure 5 shows simulation results of covering disks, suggesting a focus on the geometric arrangement and movement of the robots for coverage tasks.

31. Towards Arranging and Tightening Knots and Unknots with Fixtures:

• This paper explores robotic manipulation strategies for arranging and tightening or untying knots using external fixtures.

32. Asymptotically Optimal Feedback Planning: FMM Meets Adaptive Mesh Refinement:

• This work combines the Fast Marching Method (FMM) with adaptive mesh refinement for asymptotically optimal feedback motion planning.
• Equation (12) represents a discretization of the Hamilton-Jacobi-Bellman (HJB) equation, a fundamental equation in optimal control.
• Equation (13) and (14) show how the value function at a vertex is computed based on the values at its neighbors in the discretized space.
• Algorithm 3 outlines a “Characteristic-Driven Edge Selection” process for adaptive mesh refinement based on the value function and its dependencies.

33. Online Task Planning and Control for Aerial Robots with Fuel Constraints in Winds:

• This paper focuses on online task planning and control for aerial robots (UAVs) that have fuel limitations and operate in windy environments.
• The paper mentions a reduction to a Markov Decision Problem (MDP) for planning sequences of discrete states while minimizing fuel consumption and satisfying temporal goals specified by a Büchi automaton.
• Figure 4 illustrates optimal trajectories for visiting regions while avoiding others, demonstrating the application of their approach to a navigation task with complex temporal requirements.

Conclusion:

The collection of papers in “Algorithmic Foundations of Robotics XI” represents a snapshot of the cutting-edge research in the field. The themes of efficiency, optimality, handling uncertainty, and addressing the complexities of multi-robot systems and manipulation are central to many of the contributions. The diverse algorithmic approaches and theoretical analyses presented in these works advance the state of the art in robotic capabilities and provide a foundation for future developments.

FAQ on Algorithmic Foundations of Robotics XI

• What are some of the key challenges in multi-robot motion planning addressed in this collection of works? This collection addresses several significant challenges in multi-robot motion planning, including efficiently planning the motion of unlabeled robots (like discs) in complex environments, coordinating dynamic multi-robot tasks through coalition formation, and developing scalable approaches for large teams of robots. It also explores problems related to finding optimal paths for cooperative heterogeneous robots, and handling the complexities of task and motion planning in a unified framework.
• How are probabilistic methods and sampling-based algorithms being advanced for robot motion planning? The works presented explore various ways to improve probabilistic and sampling-based methods. This includes developing more efficient sampling strategies for optimal path planning in complex cost spaces, addressing the completeness of kinodynamic Rapidly-exploring Random Trees (RRTs), and creating real-time replanning algorithms (like RRTX) that can handle unpredictable obstacles. Furthermore, there is research on asymptotically optimal stochastic motion planning that considers temporal goals and uncertainty in the environment.
• What role does uncertainty play in the problems studied, and how is it being addressed? Uncertainty is a significant theme, appearing in areas such as robot localization with imprecise sensors, prediction of pedestrian intentions for robust avoidance, and motion planning in stochastic environments. The papers explore methods for trackability with imprecise localization, predictive modeling of uncertain intentions, and stochastic motion planning frameworks that account for state and control-dependent uncertainty, often using Gaussian belief spaces and optimization techniques.
• How are geometric and topological concepts being utilized in robot motion planning? Geometric reasoning is fundamental, with work on computing large convex regions of obstacle-free space using semidefinite programming and analyzing the complexity of arrangements in multi-robot scenarios. Topological perspectives are also explored, such as in the context of coverage algorithms for tree structures and the decidability of manipulation planning based on the topology of the robot configurations and obstacles.
• What are some of the novel algorithmic approaches being developed for specific robot types or tasks? The collection features specialized algorithms for various robotic systems and tasks. This includes efficient heuristics for combined task and motion planning, fast nearest neighbor search in the complex configuration space SE(3) relevant for many robots, planning for flexible robots like curvature and torsion constrained ribbons, and approaches for whole-body manipulation using quadratic programming. There’s also work on enabling dynamic robotic dancing through the composition of dynamical systems.
• How is the problem of multi-robot coordination and task allocation being tackled? Several papers address multi-robot coordination. One approach involves coalition formation games for dynamic tasks. Another focuses on optimal path planning in cooperative heterogeneous multi-robot delivery systems, considering both street and aerial segments. Additionally, there is work on distributed localization algorithms that allow robots to estimate their relative poses without central coordination.
• What advancements are being made in handling the interaction between robots and dynamic or unpredictable environments, including humans? The research includes strategies for real-time predictive modeling and robust avoidance of pedestrians with uncertain intentions. It also presents RRTX, a real-time motion planning algorithm designed for environments with unpredictable obstacles. These works highlight the importance of adapting plans quickly in response to changes and uncertainties in the environment.
• How are concepts from feedback control and optimization being integrated into motion planning algorithms? Optimization-based motion planning is a prominent area, with research on asymptotically optimal feedback planning that combines Fast Marching Methods (FMM) with adaptive mesh refinement. There is also work on scaling up Gaussian belief space planning through covariance-free trajectory optimization. Furthermore, the use of control barrier functions and the design of controllers for specific dynamic behaviors like robotic dancing demonstrate the integration of feedback control principles into motion planning.

Algorithmic Foundations of Robotics XI

The contents of “Algorithmic Foundations of Robotics XI” represent a cross-section of current research in robotics with a specific focus on algorithms. These algorithms draw inspiration from a variety of classical disciplines, including control theory, computational geometry and topology, geometrical and physical modeling, reasoning under uncertainty, probabilistic algorithms, game theory, and theoretical computer science. A central theme throughout the collection is the validation of algorithms, design concepts, and techniques.

The field of algorithmic foundations is particularly crucial in the current exciting time for robotics, marked by significant government initiatives and industrial investments. The increasing demand for industrial automation and the development of more capable robotic platforms necessitate the development of sophisticated algorithms. These algorithms are essential for enabling robots and automation systems to operate effectively in complex and unstructured environments. Furthermore, the applications of these algorithms extend beyond physical robotic systems to aid scientific inquiry in disciplines such as biology and neurosciences.

The research presented in this collection addresses various challenging problems within algorithmic robotics. One such problem is the coordinated motion planning of multiple bodies, specifically fully actuated, first-order point particles that need to avoid self-intersection while reaching a desired, labeled, free configuration. This is tackled using a centralized vector field planner and a hybrid controller, with a focus on computational effectiveness.

Another key area is hierarchical navigation, which involves planning motion through different levels of abstraction represented by hierarchical clustering. This includes the definition and computation of a “portal map” that serves as a dynamically computed “prepares graph” for sequentially composed particle controllers. The Hierarchical Navigation Control (HNC) Algorithm leverages hierarchy-invariant control policies and discrete transition rules in the space of binary trees to bring almost any initial configuration to a desired goal configuration without collisions.

The algorithmic foundations also encompass approaches to motion planning under uncertainty. This includes methods that deal with stochastic action uncertainty to achieve high-level tasks specified using temporal logic. Frameworks are being developed to compute optimal control policies that maximize the probability of satisfying such specifications, often by abstracting the continuous stochastic system to a discrete Markov model.

Different algorithmic paradigms are explored in the sources. Sampling-based algorithms, like Rapidly-exploring Random Trees (RRT) and Probabilistic Roadmaps (PRM), are widely used in motion planning. The collection also delves into subdivision approaches, which offer resolution-exactness by using soft predicates, providing a balance between practical implementation and theoretical guarantees. Exact algorithms represent another approach, though their implementation can be challenging due to numerical errors. Optimization-based planning is also a significant area, particularly for high degree of freedom robots, where trajectories are optimized based on various constraints such as collision avoidance, smoothness, and stability. Additionally, lattice-based motion planning algorithms are utilized for efficient computation of paths in a discretized state space, particularly in scenarios involving maximum reward collection in stochastic environments.

The algorithmic foundations of robotics also extend to multi-robot systems, addressing problems such as coordinated motion planning and multi-robot task allocation (MRTA). Coalition formation games are used to model the process of finding optimal robot coalitions based on task-related preferences.

In summary, “Algorithmic Foundations of Robotics XI” highlights the diverse and interdisciplinary nature of algorithms in robotics. It showcases research that aims to develop theoretically sound and practically effective algorithms for a wide range of challenging problems, from coordinated motion and hierarchical planning to handling uncertainty and multi-robot coordination. The emphasis on validation and the exploration of different algorithmic paradigms underscore the ongoing advancements in this critical field.

Multi-Robot Planning: Concepts and Approaches

Based on the sources and our previous discussion, multi-robot planning is a fundamental problem in robotics that involves coordinating the motions of multiple robots within a shared workspace to achieve individual or collective goals while avoiding collisions with each other and the environment. This field draws upon various algorithmic foundations, including computational geometry, graph theory, and optimization [Our conversation history].

Here’s a breakdown of multi-robot planning concepts discussed in the sources:

• Problem Definition and Complexity:
• The basic goal is to move each robot from a start to a target position without collisions.
• This is a natural extension of single-robot motion planning but is significantly more complex due to the increased number of degrees of freedom. Even for simple disc robots, the problem becomes hard when the number of robots is not constant; it has been shown to be PSPACE-hard for rectangular robots and strongly NP-hard for disc robots in a simple polygon.
• Variants of the Problem:
• The classical formulation assumes that robots are distinct and each has a specific target position.
• The unlabeled variant considers all robots to be identical and interchangeable. A generalization of this is the k-color motion-planning problem, with several groups of interchangeable robots.
• Approaches to Multi-robot Motion Planning:
• Sampling-based techniques have gained significant attention due to their relative ease of implementation and effectiveness in practice, especially for problems with many degrees of freedom. While single-robot sampling-based methods can be applied to multi-robot systems by treating the group as one composite robot, much work aims to exploit the unique properties of the multi-robot problem.
• Composite Roadmaps: One approach involves constructing a composite roadmap, which is the Cartesian product of the roadmaps of individual robots. However, the explicit construction can be computationally expensive for many robots. Implicit representations of composite roadmaps are also explored.
• Discrete RRT (dRRT): This is a pathfinding algorithm for implicitly represented geometrically embedded graphs and can be used for exploration in multi-robot motion planning on composite roadmaps. It reuses computed information to avoid costly operations like collision checking between robots and obstacles by forcing individual robots to move on pre-calculated individual roadmaps.
• Centralized vs. Decoupled Planning: Centralized planners treat all robots as a single system, while decoupled planners compute trajectories for each robot independently. Sampling-based planners can be used to compare these approaches. Optimal decoupling into sequential plans has also been proposed.
• Heuristic Methods: Due to the complexity, many heuristic methods have been developed for multi-robot task allocation (MRTA) problems, often viewed as optimal assignment problems considering individual robot constraints.
• Market-based strategies using auctions are distributed approaches for task allocation, though they can face challenges with remote robots and communication overhead.
• Coalition Formation Games: This approach models the formation of robot groups (coalitions) to perform dynamic tasks that require diverse resources. A task coordinator is responsible for forming these coalitions based on resource requirements and costs, aiming for stable coalitions where no group has a better alternative.
• Multi-robot Manipulation: Planning for multi-robot manipulation, especially in cluttered environments, is challenging because the motion of the manipulated object changes the connectivity of the robots’ free space. The Feasible Transition Graph (FTG) is a data structure that encodes object configurations based on robot free space connectivity and transitions between these configurations, providing a framework for complete multi-robot manipulation planning. This approach helps in reasoning about resource allocation, such as the number and placement of robots needed.
• Multi-robot Routing: The Multi-Robot Routing (MRR) problem focuses on efficiently utilizing a team of robots to visit multiple goal locations without preference for which robot visits which target or the order of visits. While optimal solutions are NP-hard, approximation algorithms with polynomial computational complexity have been developed, along with collision avoidance schemes to ensure robots safely reach their goals.
• Planning Under Uncertainty in Multi-robot Systems: Stochastic Extended LQR (SELQR) can be extended to plan in the belief space of multiple robots when sensing is imperfect, aiming to minimize the expected value of a cost function while considering motion uncertainty.
• Graph-based Multi-robot Path Planning: For scenarios where robots move on a graph, such as the pebble motion problem, feasibility tests and planning algorithms have been developed. For example, a group theoretic approach provides a linear-time algorithm for testing feasibility in pebble motion on graphs with rotations (PMR).

In summary, multi-robot planning is a complex and active area of research with various facets, ranging from fundamental motion planning to sophisticated task allocation and manipulation strategies, often addressing the challenges of computational complexity and uncertainty. The development of efficient and robust algorithms for coordinating multiple robots is crucial for a wide range of applications.

Fundamentals of Robot Motion Planning

The motion planning problem is a fundamental challenge in robotics that involves finding a valid (e.g., collision-free) path or trajectory for a robot to move from a start configuration to a goal configuration in its workspace. A configuration describes the robot’s pose (position and orientation) and volume occupied in the workspace. The set of all possible configurations forms the configuration space (C-space). The subset of C-space where the robot is not in collision with obstacles is called the free space (Cfree). The motion planning problem then becomes finding a continuous path within Cfree that connects the initial and goal configurations.

Here’s a more detailed discussion of the motion planning problem based on the provided sources:

• Complexity: Motion planning is generally a computationally hard problem. Designing complete planners for high-dimensional systems (more than 5 degrees of freedom) is often intractable. For multiple independent objects, the problem is PSPACE-hard. Even for a single omnidirectional point robot in a 3D environment with polyhedral obstacles, finding an optimal path is PSPACE-hard.
• Variations and Considerations:
• Static vs. Dynamic Environments: The basic problem considers static obstacles. However, many real-world scenarios involve moving obstacles, requiring continuous re-evaluation of plans to identify valid trajectories given current and predicted obstacle positions. Planning in unknown environments with obstacles having unpredictable trajectories presents additional challenges, emphasizing the importance of safety and collision avoidance.
• Kinematics and Dynamics: Motion planning can consider only the geometry (kinematics) or also the motion constraints (kinodynamics) of the robot. Kinodynamic planning seeks trajectories that satisfy both kinematic and dynamic constraints. Some work explores planning with fixed time steps in kinodynamic RRTs, noting that they might not be probabilistically complete.
• Uncertainty: In many real-world scenarios, there is uncertainty in the robot’s actions and the environment. Motion planning under uncertainty aims to find robust control strategies or policies over the state space, rather than a single trajectory, to maximize the probability of task completion despite this uncertainty. This often involves using Partially Observable Markov Decision Processes (POMDPs) or considering Gaussian belief spaces.
• Optimality: While finding a feasible path is often the primary goal, optimal motion planning seeks to find a path that minimizes a certain cost function, such as path length, time, or energy. Achieving optimality, especially for systems with dynamics, often requires specialized steering functions.
• Multi-robot Planning: As discussed in our conversation history, extending motion planning to multiple robots introduces significant complexity due to the increased degrees of freedom and the need to avoid collisions between robots in addition to static obstacles. Different approaches, such as centralized and decoupled planning, composite roadmaps, and graph-based methods, are used to tackle this problem [Our conversation history].
• Approaches to Motion Planning: The sources highlight several algorithmic approaches to address the motion planning problem:
• Sampling-based Planners: These methods, including Probabilistic Roadmaps (PRMs) and Rapidly-exploring Random Trees (RRTs), build an approximate representation of the free space by randomly sampling configurations and connecting them to form a graph or tree. While effective in many high-dimensional problems, they can struggle with narrow passages and may not guarantee optimality. Variants like **RRT* ** aim for asymptotic optimality. RRT-connect is an efficient approach for single-query path planning. MRdRRT adapts RRT for multi-robot motion planning on implicitly represented composite roadmaps.
• Optimization-based Planners: These methods formulate motion planning as an optimization problem, where a trajectory is computed by minimizing a cost function subject to various constraints like collision avoidance and smoothness. Examples include using potential fields, elastic strips/bands, and direct encoding of constraints into optimization costs solved with numerical solvers. Stochastic Extended LQR (SELQR) is used for optimization-based planning under uncertainty. Asymptotically Optimal Feedback Planning combines the Fast Marching Method with adaptive mesh refinement to compute optimal feedback plans.
• Exact Algorithms: These algorithms aim to find a solution if one exists or report that none exists, often by explicitly constructing the free space or its connectivity. However, they can be computationally very expensive, especially for higher degrees of freedom.
• Subdivision Approaches: These methods, like the one presented for link robots, use soft predicates and recursive subdivision of the configuration space to achieve resolution-exactness, balancing practicality and theoretical guarantees.
• Heuristic Methods: Many problems, especially in dynamic or multi-robot settings, rely on heuristic approaches to find solutions efficiently, even if completeness or optimality cannot be guaranteed. FFRob extends the heuristic ideas from symbolic task planning to motion planning.
• Graph-based Planning: In some cases, the motion planning problem can be abstracted to finding a path on a graph, for example, in the pebble motion problem. Efficient algorithms exist for testing feasibility and finding plans for such problems, sometimes considering rotations of the pebbles.
• Reactive Planning: These approaches focus on quickly reacting to changes in the environment, often using local planning methods like Artificial Potential Fields (APF).
• Human-Assisted Planning: Recognizing the strengths of human intuition for high-level scene analysis and the machine’s precision for low-level tasks, collaborative planning strategies like Region Steering allow users to guide sampling-based planners.
• Integration with Task Planning: For more complex robotic tasks, motion planning is often integrated with high-level task planning. This involves coordinating symbolic reasoning about the sequence of actions with geometric planning of the robot’s movements.

In conclusion, the motion planning problem is a multifaceted challenge in robotics with significant theoretical and practical implications. The choice of approach depends on the specific requirements of the task, the complexity of the robot and environment, and the need for completeness, optimality, and robustness in the presence of uncertainty and dynamic changes. The research highlighted in the sources continues to advance the algorithmic foundations of motion planning, addressing its various complexities and striving for more efficient, reliable, and adaptable solutions.

Robot Collision Avoidance Strategies

Collision avoidance is a fundamental aspect of motion planning, ensuring that a robot can move from a start to a goal configuration without coming into contact with obstacles in the environment or with other robots. The sources provide several insights into different approaches and considerations for collision avoidance in various scenarios.

Here’s a discussion of collision avoidance drawing from the provided material:

• Core Requirement: A primary goal of motion planning is to find a path or trajectory that is collision-free. This means that at no point in time along the planned motion should the robot’s physical extent overlap with any part of the obstacle space.
• Configuration Space (C-space): The concept of C-space is central to collision avoidance. The obstacle space (Cobst) represents all configurations where the robot is in collision, and the goal is to find a path within the free space (Cfree), which is the set of collision-free configurations.
• Types of Obstacles: Collision avoidance needs to consider different types of obstacles:
• Static Obstacles: These are fixed in the robot’s workspace. Most traditional motion planning algorithms inherently address avoidance of these by ensuring the planned path stays within Cfree.
• Dynamic Obstacles: These are obstacles whose position changes over time. Avoiding these requires predicting their future positions and velocities and planning accordingly.
• Other Robots: In multi-robot systems, robots must avoid collisions not only with the environment but also with each other.
• Single Robot and Dynamic Obstacles: Several techniques are discussed for avoiding collisions with moving obstacles:
• Collision Prediction: A novel geometric method is proposed to predict collisions with rigid and articulated obstacles with unknown motion. This approach models obstacles as adversarial agents that will move to minimize the time the robot remains collision-free. The Earliest Collision Time (ECT) is calculated to determine how long the robot can safely follow its current path before a potential collision. This allows for adaptive replanning when a critical collision time is approaching, rather than replanning at fixed intervals. This method can handle arbitrary polygon shapes and articulated objects, overcoming limitations of methods that assume simpler geometries like discs.
• Stochastic Reachable (SR) Sets: These sets are used to determine collision avoidance probabilities in dynamic environments with uncertain obstacle motion. By formulating a stochastic reachability problem, the probability of avoiding collision can be calculated. Integrating SR sets with Artificial Potential Fields (APF-SR) has shown high success rates in avoiding multiple moving obstacles by using the likelihood of collision to construct repulsion fields.
• Inevitable Collision States (ICS) and Velocity Obstacles (VO): These are existing concepts where ICS represent states from which collision is unavoidable, and VO are sets of velocities that would lead to collision. These methods often require some information or assumptions about the future motion of the obstacles.
• Multiple Robot Collision Avoidance: Planning for multiple robots adds significant complexity:
• Increased Degrees of Freedom: Treating multiple robots as a single system increases the dimensionality of the configuration space.
• Centralized vs. Decoupled Approaches:Centralized planners consider all robots together, but their complexity grows rapidly with the number of robots.
• Decoupled planners plan paths for each robot independently and then try to coordinate them to avoid inter-robot collisions.
• Reciprocal Collision Avoidance (RVO) and Optimal Reciprocal Collision Avoidance (ORCA): These are popular decoupled approaches where each robot computes velocities to avoid collisions, assuming other robots will also react to avoid collisions. ORCA defines permissible velocities as half-planes, leading to smooth and oscillation-free motion. Acceleration-Velocity Obstacles (AVO) extend this by considering acceleration limits.
• Motion Graphs: For multi-robot planning with unit discs, motion graphs can represent adjacencies between start and target configurations within a connected component of the free space, ensuring collision-free movement between these configurations. The concept of collision discs (D2(x)) defines the area around a robot where another robot cannot be without collision.
• Composite Roadmaps: For multiple robots, individual Probabilistic Roadmaps (PRMs) can be combined into a composite roadmap (e.g., using a tensor product). This allows for querying collision-free paths for the entire group of robots, and pre-computed individual roadmaps can reduce the need for repeated collision checks with static obstacles.
• Well-Separated Configurations: Some problem formulations assume that start and target configurations of robots are “well-separated” to simplify initial and final collision avoidance.
• Human Assistance: In some approaches, humans can aid collision avoidance by providing high-level guidance and identifying regions to avoid, allowing the automated planner to handle the detailed collision checking and pathfinding.
• Collision Avoidance in Manipulation: When a robot manipulates movable objects, collision avoidance must consider the robot, the object being manipulated, and the environment. This can involve maintaining contact while avoiding collisions.
• Geometric Representation and Collision Checking: Efficient collision detection algorithms and geometric representations of robots and obstacles (e.g., bounding boxes, collision discs, polygons) are crucial for the practical implementation of collision avoidance strategies.
• Smoothness and Stability: Collision avoidance is often coupled with the desire for smooth and dynamically stable robot motions, especially for high-DOF robots. Optimization-based methods often incorporate smoothness and stability constraints alongside collision avoidance.

In summary, collision avoidance is a central challenge in motion planning that requires careful consideration of the environment’s dynamics, the number and complexity of robots, and the desired properties of the resulting motion. Various algorithmic approaches have been developed, each with its strengths and limitations in addressing different collision avoidance scenarios.

Probabilistic Completeness in Sampling-Based Motion Planning

Probabilistic completeness is a crucial property for sampling-based motion planning algorithms. It essentially means that if a solution to a motion planning problem exists, the probability that the algorithm finds it approaches one as the running time (or number of samples) tends to infinity. The sources discuss probabilistic completeness in the context of several different motion planning algorithms:

• Rapidly-Exploring Random Trees (RRTs) and Variants:
• Standard RRTs are often considered to be probabilistically complete. However, the sources highlight that this depends on the implementation details, particularly how the tree is extended.
• It has been shown that an RRT using a fixed time step and a randomly selected input (from a finite input set U) is probabilistically complete. However, this variant is often less efficient.
• The more common variant of kinodynamic RRTs that uses a fixed time step and chooses the best control input to get as close as possible to the sampled state according to a distance metric is not generally probabilistically complete. The provided proof uses a counterexample to demonstrate this. This contradicts the general perception that all RRTs are inherently probabilistically complete.
• T-RRT and RRT*: Both T-RRT (Transition-based RRT) and RRT* are probabilistically complete.
• T-RRT*, which integrates the transition test of T-RRT into RRT*, is also probabilistically complete. This is attributed to the probabilistic completeness of RRT*, despite the non-uniform sampling due to the transition test, as the probability of a sample being accepted is never zero.
• AT-RRT (Anytime T-RRT), an extension of T-RRT, is also probabilistically complete because it behaves like T-RRT before a solution is found.
• Region Steering: This planning approach is probabilistically complete because it retains the entire workspace as an attract region, assuming that the underlying sampler it uses is also probabilistically complete. If the underlying sampler guarantees asymptotically complete coverage of the space, then Region Steering maintains this property.
• dRRT (Dynamic RRT for implicit roadmaps): This algorithm is shown to possess a strong property of probabilistically revealing all vertices of the traversed graph (if connected) with high probability, assuming the vertices are in general position. The proof relies on the fact that the random sample needs to fall within the intersection of Voronoi cells to extend the tree, and this intersection has a non-zero measure under the general position assumption.
• MRdRRT (Multi-robot dRRT): The probabilistic completeness of this multi-robot approach depends on the probabilistic completeness of the underlying single-robot roadmaps and the graph search algorithm (dRRT). While the composite roadmap approach is generally probabilistically complete with a complete graph search, in this case, the graph search (dRRT) is only probabilistically complete, requiring potential refinements to the proof as the Voronoi cell sizes tend to zero. The authors also note that dRRT can be modified to be complete for a finite composite roadmap by systematically exploring unexposed edges.
• STABLE SPARSE RRT (SST and SST*):
• SST is proven to be probabilistically δ-robustly complete under the condition that δv + 2δs < δ, where δ relates to the clearance from obstacles, and δs and δv are parameters of the algorithm. This is a weaker form of probabilistic completeness that incorporates a clearance value. The proof involves constructing a sequence of balls covering a δ-robust optimal path and showing that the algorithm has a non-zero probability of making progress along this sequence.
• SST* is an asymptotically optimal variant of SST that uses a schedule to shrink its parameters over time. It can be proven that SST* is probabilistically complete and asymptotically optimal.
• Sampling-Based Planners for Temporal Logic: While these methods can quickly find satisfying trajectories for tasks specified in Linear Temporal Logic (LTL), the source notes that they are not correct-by-construction. However, the probabilistic completeness of many sampling-based planners guarantees that if a satisfying trajectory exists, the probability of finding one grows to 1 over time.

In summary, probabilistic completeness is a desirable property for motion planning algorithms, especially those that rely on sampling. It provides a theoretical guarantee that the algorithm will eventually find a solution if one exists. However, as highlighted by the discussion on kinodynamic RRTs, achieving probabilistic completeness often depends on specific implementation choices and assumptions about the problem and the algorithm’s components. Some algorithms, like SST, offer a δ-robust form of completeness that considers clearance, while others, like SST*, can achieve both probabilistic completeness and asymptotic optimality.

Partially Admissible Clusters in Hierarchical Particle Systems

Based on the sources, a partially admissible cluster for a given configuration is defined as follows:

Definition 3: Let $x \in ( \mathbb{R}^d )^J$, $\tau \in BT_J$ and $K \subseteq J$. Then cluster $I$ of $\tau$ is said to be partially admissible for $x|K$ if $\eta_{i,I,\tau}(x) \leq 0$ for all $i \in I \cap K$.

To understand this definition, let’s break it down:

• $x \in ( \mathbb{R}^d )^J$: This represents a configuration of $J$ distinct particles in a $d$-dimensional Euclidean space.
• $\tau \in BT_J$: This denotes a rooted non-degenerate (binary) tree over the index set $J$, which represents a cluster hierarchy of the particles.
• $K \subseteq J$: This is a subset of the indices of the particles.
• $I$ of $\tau$: This refers to a cluster within the hierarchical clustering represented by the tree $\tau$. A cluster is defined as the set of leaves (particles) reachable from a vertex in the tree.
• $\eta_{i,I,\tau}(x)$: This is a scalar-valued “separation” function that depends on the configuration $x$, the cluster $I$ in the hierarchy $\tau$, and the individual particle $i$. It is defined in Equation (8) of the source as $\eta_{i,I,\tau}(x) : = ( x_i – m_{I,\tau}(x) )^T s_{I,\tau}(x)$, where $m_{I,\tau}(x)$ is the midpoint between the centroids of cluster $I$ and its local complement $I^{-\tau}$, and $s_{I,\tau}(x)$ is the separation vector between these centroids.
• $x|K$: This likely refers to the partial configuration of $x$ restricted to the particles with indices in the set $K$.

Therefore, a cluster $I$ of a hierarchy $\tau$ is partially admissible for a configuration $x$ with respect to a subset of particles $K$ if the value of the separation function $\eta_{i,I,\tau}(x)$ is less than or equal to zero for all particles $i$ that are members of both the cluster $I$ and the subset $K$.

It is also noted that for a partition ${I_\alpha}$ of a cluster $I \in C(\tau)$, the cluster $I$ of $\tau$ is admissible for $x$ if and only if $I$ is partially admissible for all $x|I_\alpha$’s. This highlights that full admissibility can be seen as a collection of partial admissibilities over the entire cluster.

Configuration Space Strata and Hierarchical Clustering

Based on the sources, a stratum in configuration space is defined as follows:

• Given a hierarchical clustering (HC), which is a relation between the configuration space and the space of binary hierarchies ($BT_J$), a stratum is associated with a specific binary hierarchy $\tau \in BT_J$.
• The stratum $S(\tau)$ is the set of all configurations $x$ in the configuration space ($Conf( \mathbb{R}^d , J )$) that support the same binary hierarchy $\tau$ under the given hierarchical clustering relation HC. In other words, $(x, \tau) \in HC$ for all $x \in S(\tau)$.
• The collection of all strata for all possible binary hierarchies forms a tree-indexed cover of the configuration space. This means that every valid configuration belongs to at least one stratum.
• For the specific HC2-means divisive hierarchical clustering method, the source defines open strata ($S^o(\tau)$) and closed strata ($S(\tau)$) based on the intersection of inverse images of a scalar-valued “separation” function $\eta_{i,I,\tau}$. These functions relate to the separation of clusters within the hierarchy for a given configuration.
• A hierarchy-invariant vector field is defined as one that, when applied to a configuration within a stratum $S(\tau)$, keeps the resulting flow within the open stratum $S^o(\tau)$ for any positive time $t$.

In essence, a stratum groups together all the robot configurations that are classified as having the same hierarchical clustering structure according to a chosen clustering method.

Hierarchy-Invariant Vector Fields and Configuration Space Stratification

Based on the sources, the defining property that characterizes hierarchy-invariant vector fields is their behavior with respect to the strata of a hierarchical clustering. Specifically:

• A vector field $f$ defined over the configuration space $Conf( \mathbb{R}^d , J )$ belongs to the class of hierarchy-invariant vector fields associated with a binary hierarchy $\tau \in BT_J$, denoted as $FHC(\tau)$, if the flow $\phi^t$ induced by $f$ on the configuration space satisfies the following condition: $\phi^t ( S(\tau) ) \subset \stackrel{\circ}{S}(\tau) , \forall t > 0$.

This means that if a system’s configuration $x$ starts within a stratum $S(\tau)$ (the set of all configurations supporting the hierarchy $\tau$), then for any positive time $t$, the configuration $\phi^t(x)$ under the flow of the hierarchy-invariant vector field $f$ will remain within the interior of the same stratum, $\stackrel{\circ}{S}(\tau)$. In other words, hierarchy-invariant vector fields keep the system’s configuration strictly within the interior of the stratum corresponding to the initial hierarchy.

Furthermore, the source notes that any hierarchy-invariant field $f \in FHC(\tau)$ must also leave the collision-free configuration space $Conf( \mathbb{R}^d , J )$ invariant, thus avoiding any self-collisions of the particles along the resulting trajectories.

Configuration Space Strata in Hierarchical Robot Clustering

Based on the sources, a stratum in configuration space is a fundamental concept related to hierarchical clustering of robot configurations. Here’s a brief explanation:

• Given a hierarchical clustering (HC), which defines a relationship between the space of robot configurations ($Conf( \mathbb{R}^d , J )$) and the abstract space of binary hierarchies ($BT_J$), a stratum is associated with a specific binary hierarchy $\tau$ from $BT_J$.
• The stratum $S(\tau)$ is formally defined as the set of all configurations $x$ within the configuration space that support the same binary hierarchy $\tau$ according to the hierarchical clustering relation HC. Mathematically, $S(\tau) : = { x \in Conf ( \mathbb{R}^d , J ) \mid (x, \tau ) \in HC }$.
• The collection of all these strata, indexed by the different possible binary hierarchies, forms a tree-indexed cover of the configuration space. This implies that every valid configuration of the robots will belong to at least one such stratum.
• For the specific HC2-means divisive hierarchical clustering method, strata can be characterized using a separation function $\eta_{i,I,\tau}(x)$. Open strata ($S^o(\tau)$) and closed strata ($S(\tau)$) are defined based on the values of this function for all clusters $I$ in the hierarchy $\tau$. A configuration $x$ belongs to $S(\tau)$ if and only if every cluster $I$ of $\tau$ is admissible for $x$, meaning $\eta_{i,I,\tau}(x) \leq 0$ for all $i \in I$.
• As discussed in our conversation history, hierarchy-invariant vector fields are characterized by the property that they keep the system’s configuration within the interior of the stratum corresponding to the initial hierarchy.

In essence, a stratum groups together all the different arrangements of the robots that share the same underlying hierarchical clustering structure as determined by a chosen hierarchical clustering method. These strata provide a way to decompose and analyze the complex configuration space.

Robot Configuration and Obstacle Space

Based on the sources, the relationship between obs(x) and the obstacle space O is as follows:

• The obstacle space O is defined as the region in $R^2$ that is outside of the robot’s workspace $W$. It represents the areas that the robot should avoid.
• For a given configuration $x \in W$ (where the robot’s center is at point $x$), the obstacle set of $x$, denoted as obs(x), is defined as the set of all points $y$ that satisfy two conditions:
• $y$ belongs to the obstacle space O: $y \in O$.
• The Euclidean distance between $x$ and $y$ is less than 1: $|x – y| < 1$. This means that the point $y$ in the obstacle space is within the open unit disc $D_1(x)$ centered at $x$.
• In other words, obs(x) is the portion of the obstacle space O that overlaps with the open unit disc $D_1(x)$ centered at the robot’s center $x$. Since the robots are considered unit discs, obs(x) contains the points in the obstacle space that the robot would collide with if its center were at $x$.
• It is also noted that obs(x) is empty ($\emptyset$) if and only if the configuration $x$ is in the free space $F$. The free space $F$ is the set of all configurations $x$ where the open unit disc centered at $x$ does not intersect the obstacle space $O$ ($D_1(x) \cap O = \emptyset$). If $x$ is in $F$, then there are no points in $O$ within a distance of 1 from $x$.

Therefore, obs(x) is a subset of the obstacle space O that specifically identifies the obstacles in immediate proximity to a robot centered at configuration x (those within a distance of 1). This concept is crucial for defining collisions between the robot and the obstacle space.

Robot Collision Disc Definition and Significance

Based on the sources, the collision disc of a configuration is defined as follows:

For any configuration $x \in R^2$ (representing the location of the center of a robot) and $r \in R^+$, $D_r(x)$ is defined as the open disc of radius $r$ centered at $x$.

Specifically, the collision disc of the configuration $x$, denoted as $D_2(x)$, is the open disc of radius 2 centered at $x$.

The significance of the collision disc lies in defining collisions between robots: if a robot is at configuration $x$, then no other robot can be at a configuration $y$ within the interior of $D_2(x)$, denoted as $Int(D_2(x))$, as this would result in a collision. This is because the robots themselves are considered to be open unit discs. Two unit discs collide if and only if the distance between their centers is less than $1 + 1 = 2$. Therefore, $D_2(x)$ represents the region where the center of another robot cannot be located without colliding with a robot centered at $x$.

Furthermore, the source introduces a related concept, $D^*(x)$, which is the part of the collision disc $D_2(x)$ that is within the same free-space component $F_i$ as $x$. That is, $D^*(x) := D_2(x) \cap F_i$, where $x \in F_i$.

Well-Separated Robot Configurations: Definition and Implications

Based on the sources, the condition for well-separated start and target positions is explicitly defined in. Given sets of start configurations $S = {s_1, s_2, …, s_m}$ and target configurations $T = {t_1, t_2, …, t_m}$, where $S, T \subset F$ (the free space), the configurations in $S$ and $T$ are assumed to be well-separated if the following condition holds:

For any two distinct configurations $x, y \in S \cup T$, we have $|x – y| \geq 4$.

This condition states that the Euclidean distance between the centers of any two different robots at their start positions, any two different robots at their target positions, or any start and any target position must be greater than or equal to 4.

Considering that the robots are unit discs (radius 1), this separation distance of 4 ensures that the collision discs of any two distinct start or target configurations do not overlap. Recall from our conversation that the collision disc of a configuration $x$, $D_2(x)$, is an open disc of radius 2 centered at $x$. If $|x – y| \geq 4$, then the distance between the centers of the two collision discs $D_2(x)$ and $D_2(y)$ is at least 4, which is greater than the sum of their radii ($2 + 2 = 4$). Therefore, $D_2(x) \cap D_2(y) = \emptyset$ for any two distinct $x, y \in S \cup T$.

Furthermore, Lemma 2 in the source explicitly mentions that for any two distinct configurations $x_1, x_2 \in S_i \cup T_i$ (start and target positions within a component $F_i$), $D^*(x_1) \cap D^*(x_2) = \emptyset$, because $D_2(x_1) \cap D_2(x_2) = \emptyset$ due to the assumption that the start and target positions are well-separated. Here, $D^*(x) = D_2(x) \cap F_i$ is the part of the collision disc within the same free-space component as $x$.

The paper notes in that while a separation distance of 4 ensures that a solution always exists (given that each connected component contains the same number of start and target positions), this might not be the case with a smaller separation distance, as illustrated in Figure 4. In the example, even with the same number of start and target positions in the free space, a smaller separation prevents the robots from simultaneously reaching their goals because one robot blocks the other’s path.

K-Color Multi-Robot Motion Planning: An Overview

Based on the sources, the k-color motion-planning problem is a generalization of the unlabeled multi-robot motion planning problem.

In this variant, instead of having a single group of completely identical and interchangeable robots (as in the standard unlabeled problem), there are several distinct groups of interchangeable robots, where each group can be considered a different “color”.

Therefore, the key characteristic of the k-color motion-planning problem is the presence of k different groups (colors) of robots, where robots within the same group are interchangeable, but robots from different groups are distinguishable (by their color) and are not interchangeable with robots from other groups.

The source cites as the origin of this problem proposal.

Hierarchical Navigation Control Algorithm for Multi-Particle Systems

The core idea of the Hierarchical Navigation Control (HNC) Algorithm is to solve the centralized motion planning problem for multiple distinct Euclidean particles by using a hierarchical decomposition of the configuration space based on the HC2-means clustering method. It achieves this by combining continuous control within individual strata with discrete transitions between adjacent strata in a provably correct and computationally efficient manner.

Here’s a breakdown of the key aspects of the HNC Algorithm:

• Hierarchical Decomposition of Configuration Space: The algorithm utilizes the concept of strata, where each stratum $S(\tau)$ corresponds to a specific binary hierarchy $\tau$ of the particles obtained through HC2-means clustering. The entire configuration space is covered by these tree-indexed strata [as implied by previous conversations].
• Intra-Stratum Navigation (Hybrid Base Case): When the current configuration $x$ belongs to the same stratum $S(\tau)$ as the desired goal configuration $y$ (which supports $\tau$, $y \in \stackrel{\circ}{S}(\tau)$), the algorithm applies a stratum-invariant continuous controller $f_{\tau,y}$ (from Algorithm 1 in). This controller, based on hierarchy-invariant vector fields, ensures that the system stays within $S(\tau)$ and asymptotically approaches the goal $y$ without collisions. This is treated as the “hybrid base case”.
• Inter-Stratum Navigation (Hybrid Recursive Step): If the current configuration $x$ (supporting hierarchy $\sigma$) is not in the same stratum as the goal $y$ (supporting hierarchy $\tau$), the algorithm enters a “hybrid recursive step” to navigate across strata. This involves:
• Discrete Transition in Hierarchy Space: Invoking the NNI (Nearest Neighbor Interchange) transition rule $g_{\tau}$ (from Algorithm 2 in) on the space of binary trees $BT_J$. This rule proposes an adjacent hierarchy $\gamma$ in the NNI-graph that is closer to the goal hierarchy $\tau$ in terms of a discrete Lyapunov function. The NNI-graph $N_J$ is a subgraph of the adjacency graph $A_J$ of the HC2-means hierarchies.
• Defining a Local Goal using the Portal Map: Choosing a local configuration goal $z$ within the portal between the current stratum $S(\sigma)$ and the proposed adjacent stratum $S(\gamma)$. This local goal $z$ is computed using the portal map $Port(\sigma, \gamma)(x)$. The portal $Portal(\sigma, \tau) = \stackrel{\circ}{S}(\sigma) \cap \stackrel{\circ}{S}(\tau)$ represents the set of configurations supporting interior strata of both hierarchies. The portal map provides a computationally effective geometric realization of the edges of the NNI-graph in the configuration space. It retracts $S(\sigma)$ into the set of standard portal configurations in $Portal(\sigma, \gamma)$.
• Continuous Control Towards the Local Goal: Applying another stratum-invariant continuous controller $f_{\sigma,z}$ (from Algorithm 1) to drive the system from the current configuration $x$ within $S(\sigma)$ towards the local goal $z \in Portal(\sigma, \gamma)$. This ensures the state remains within $S(\sigma)$ during this phase.
• Transitioning to the Next Stratum: Once the trajectory reaches a sufficiently small neighborhood of $z$ (and hence enters $Portal(\sigma, \gamma) \subset S(\gamma)$ in finite time), the algorithm updates the current hierarchy to $\sigma \leftarrow \gamma$ and repeats the recursive step (2a) until the configuration enters the goal stratum $S(\tau)$, at which point the base case (step 1) is applied.
• Hybrid Dynamical System: The HNC Algorithm defines a hybrid dynamical system by alternating between discrete transitions in the space of hierarchies (using the NNI-graph) and continuous motion within the strata (using hierarchy-invariant vector fields and the portal map).
• Correctness and Efficiency: The algorithm guarantees that almost every initial configuration will reach an arbitrarily small neighborhood of the desired goal configuration $y$ in finite time, without any collisions along the way. Each discrete transition and the computation of the portal location can be done in linear time, $O(|J|)$, with respect to the number of particles $|J|$. The NNI transition rule $g_{\tau}$ ensures progress towards the goal hierarchy $\tau$ by reducing a discrete Lyapunov function.

In summary, the HNC Algorithm’s core idea is to systematically navigate through the configuration space by moving within well-defined strata using continuous, collision-avoiding control, and transitioning between adjacent strata (that are closer in the hierarchical clustering space) using a discrete process guided by the NNI-graph and geometrically realized by the portal map. This hybrid approach provides a computationally effective and provably correct method for multi-particle motion planning.

By Amjad Izhar
Contact: amjad.izhar@gmail.com
https://amjadizhar.blog

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