Asymptotically Optimal Samplingbased Planners
1 Synonyms
AO planning, optimal motion planning
2 Definition
An asymptotically optimal samplingbased planner employs sampling to solve robot motion planning problems and returns paths with a cost that converges to the optimal solution cost, as the number of samples approaches infinity.
3 Overview
Samplingbased motion planners sample feasible configurations of a robot and connect such samples with paths that are valid for the robot to traverse. They are widely popular in robotics due to their simplicity, generality, and elegance, which allows for analysis of their properties. They scale well to highdimensional motion planning problems and rely only on simple, wellunderstood primitives, such as collision checking and nearest neighbor datastructures.
Some of these planners, such as the Probabilistic Roadmap Method (PRM) (Kavraki et al 1996), construct a graph, where nodes correspond to configurations and edges correspond to local paths. The graph is referred to as a roadmap, which can be preprocessed, and can then be used to effectively answer multiple queries. Other alteratives, such as the Rapidly Exploring Random Tree (RRT) (LaValle and Kuffner Jr 2001) algorithm, build a tree data structure rooted at an initial configuration and aim to quickly explore the reachable configuration space for solving a specific query. The treebased variants can also be used in motion planning challenges that involve significant dynamics, where there may not be access to a method (i.e., a steering function) for connecting two robot configurations with a local path.
Many of the early samplingbased planners aimed to provide the property of probabilistic completeness (Hsu et al 1999) This property guarantees that a solution to a motion planning problem will be found, if one exists, as the number of configuration space samples approaches infinity. Further analysis focused on properties of configuration spaces, which allowed samplingbased planners to be effective, such as goodness (Kavraki et al 1998) and expansiveness (Hsu et al 1999). Many variants were proposed, which focused on enhancing the practical performance of samplingbased planning (Amato et al 1998; Kim et al 2003; Raveh et al 2011).
Given the hardness of the motion planning problem (Canny 1988), the early methods did not focus on returning paths of highquality. As the field matured and the planners became increasingly faster, however, the focus transitioned towards understanding the conditions under which these methods can asymptotically converge to paths that optimize a desirable cost function. This quest gave rise to a new family of samplingbased planners, which can also achieve asymptotic optimality.
Mathematical Formulation
Consider a robot in a workspace with obstacles, where a configuration is a vector of robot variables, which fully define the volume occupied by the robot in the workspace.
Definition 1 (Configuration Space)
All possible configurations of the robot define a set . The feasible subset and the infeasible subset refer to configurations that do not result or cause collisions with obstacles in the workspace, respectively.
The notion of collisions can be generalized to express any feasibility constraint.
Definition 2 (Feasible Paths with Strong clearance)
A feasible path is a parameterized continuous curve of bounded variation. The set of all possible such paths is . A feasible path has strong clearance, if lies entirely inside the interior of , i.e., .
The clearance property guarantees there is always a dimensional ball in  and, thus, a positive volume  around configurations of a solution path.
Definition 3 (Robustly Feasible Motion Planning)
Given and a set , the robustly feasible motion planning problem asks for a feasible path with strong clearance, so that and .
Definition 4 (Optimal Motion Planning)
Given a path cost function , an optimal solution to the motion planning problem satisfies: .
The optimal solution need not be unique but the minimum cost is unique and finite. Let define the extended random variable corresponding to the cost of the minimumcost solution returned by algorithm after iterations.
Definition 5 (Asymptotic Optimality)
An algorithm is asymptotically optimal if, for a robustly feasible motion planning problem , which admits a robustly optimal solution with finite cost ensures that:
Existing literature provides the guarantees of asymptotic optimality in terms of this event occurring asymptotically almost surely (Karaman and Frazzoli 2011) or in probability (Janson et al 2015). Asymptotically almost sure convergence implies that after a large enough number of samples, the event of the solution cost converging to the optimal cost occurs infinitely often. Convergence in probability says that the probability of the solution cost being equal to the optimal cost is 1 at infinite samples. This is highlighted in Fig 1.
4 Key Research Findings
Identifying the conditions for samplingbased motion planners to achieve asymptotic optimality was first achieved (Karaman and Frazzoli 2011) by building on top of results in random geometric graphs (Penrose 2003) and resulted in a series of new algorithms that provide this property, such as PRM and RRT (Karaman and Frazzoli 2011).
Outline of PRM and RRT: Algo 1 describes the PRM algorithm. The method samples configurations in and proceeds to iterate over each of them and add valid edges within a neighborhood of each vertex, in order to generate a roadmap . The query phase consists of connecting the starts and goals to the roadmap, and then performing graph search () to return a solution path. In practice the same roadmap can be used for multiple queries or pairs of starts and goals.
Algo 2 refers to the RRT algorithm. For every query this incremental approach constructs a tree rooted at the initial configuration. At any iteration, the nearest tree node from a sample is used to steer towards the sample to generate a candidate node to add. For each such candidate node , the best parent is selected from the valid neighborhood of . The best connections are decided in terms of the costtogo from the root of the tree and the cost of an edge connecting two configurations . The neighborhood is also rewired by checking connections from the added node to the neighbors. This ensures that the algorithm maintains the optimal tree from the generated nodes. The solution can be returned by tracing the tree nodes from a discovered goal to the root.
Note: The original work also described an RRG algorithm that broadly operates like RRT, but keeps all the edges in each neighborhood in the subroutine.
As demonstrated by the Figure 2, both these algorithms attempt to discover the solution path over connections that exist on an underlying graph comprising of configurations in . The specification of a neighborhood heavily affects the solutions returned. If the properties of the underlying graph generated from random samples using the corresponding neighborhoods can be described, it might be possible to study path quality guarantees over these graphs.
(Bottom): Different steps of the RRT algorithm are highlighted. (Left): The dashed circle represents the random sample, the nearest node of the existing tree, and the result(blue point v) of steering towards the random sample. (Middle): The neighborhood of vis then checked for the best connections. (Right): The solution is traced over this tree .
Conditions for Asymptotic Optimality
The searchstructure generated by an AO algorithm can be modeled as a random geometric graph (RGG) in the of a motion planning problem.
Definition 6 (Random Geometric Graph for Motion Planning)
The vertices of a random geometric graph correspond to i.i.d. uniformly sampled configurations in . Each vertex is connected with edges that lie in to all configurations that are within a distance away.
The related literature provides a comprehensive description of the properties of such graphs (Penrose 2003). Specifically, the percolation property of a random geometric graphs refers to the condition under which the largest connected component of the graph grows infinitely large as grows to infinity. In other words, when such a graph percolates, it fills up the space in which the vertices are sampled. In terms of connectivity, is surely connected when
and disconnected otherwise, where is the volume of a dimensional ball of radius 1, and is constructed in a dimensional unit hypercube. This connectivity threshold extends to asymptotic optimality requirements in motion planning under conditions that ensure graph connectivity in the vicinity of the optimal path .
Outline of the Analysis: Leveraging the clearance of any feasible solution (Definition 2) returned by the algorithm, there exists a volume of surrounding such a . Typically, the optimal path itself can touch the boundary of and hence possess 0clearance at such contact points. However, as long as there exists a sequence of paths, each having some clearance for , such that , the algorithm can operate over a positive volume around each such . This positive volume is necessary to maintain nondegenerate probability of sampling in these regions. This sequence of is called a sequence of observing paths. In general, as increases, , and decrease, meaning practically demonstrates the ability of the algorithm to discover solutions through increasingly narrower corridors of as increases.
Each has a volume of described by the clearance around it. The volume surrounding an observing path is divided into a finite sequence of tiling constructions (hyperballs in Karaman and Frazzoli (2011); Janson et al (2015) and hypercubes in Solovey et al (2018)), where each tile has a positive volume.
By setting an appropriate value, the algorithm has to ensure connectivity in the clearance volume around each observing path , to return a solution path along consecutively connected configurations along .
knear variants: There are variations of random geometric graphs, where each vertex is connected with its closest neighbors as long as the corresponding edge again lies in . The algorithms can follow either description of the neighborhood as a function of the number of samples ( or )
The knear analysis broadly operates over the expected number of samples in the volumes described by the variants and appropriate tiling constructions.
The lower bounds for the connection radius and for the resulting
methods (Karaman and Frazzoli 2011) are as follows:
PRM:  
kPRM: 
where represents the dimensional volume measure, is the natural logarithm, and is Euler’s number.
Note on treebased planners: Recent work (Solovey et al 2019) argues that treebased AO samplingbased motion planners need to account for the existence of a chain of samples from the start along the hyperball tiling, for every hyperball, in addition to ensuring that the hyperball has a sample in it with increasing . This leads to a connection ratio bound that is for treebased planners.
Alternative Analysis Models
After the initial foray into studying asymptotic optimality properties of samplingbased planners (Karaman and Frazzoli 2011), alternative  and in some cases simpler  models were developed for the analysis of these planers, which go beyond the foundations of random geometric graphs.
Batched and Deterministic Sampling: A line of research has focused on sampling sequences towards extending the random geometric graph model (Janson et al 2015)(Janson et al 2018). In particular, in the context of analyzing the FMT planner (Janson et al 2015), it was possible to achieve slightly relaxed theoretical guarantees for convergence (in probability) for uniform sampling sequences. The algorithm operates rather similar to a batchversion of PRM but uses a slightly smaller connection radius of
The underlying analysis enables to deduce the convergence rate bound for PRM and FMT as , when the algorithm is executed for samples in a dimensional configuration space, where is some arbitrarily small constant. The convergence rate of AO samplingbased algorithms is an important factor, which dominates the finite time properties and practical performance of solutions.
Subsequent work demonstrates the use of lowdispersion sampling sequences (Sukharev, and Halton), which guarantee points with a dispersion of (Janson et al 2018), where is the set of nodes of the samplingbased planner. This is tighter than the expected dispersion from uniform sampling, and results in smaller bounds for the necessary connection radius for AO, where the algorithm is AO if . The corresponding work provides a set of relations between dispersion and convergence rates as well as computational complexity results.
AO Monte Carlo Trees: To build a graph in , it is necessary to connect two nodes with a local path via a steering function. This is not always feasible for systems with dynamics, as it requires solving an ordinary differential equation with boundary constraints, i.e., a boundary value problem. Thus, the results described so far do not apply to systems with significant dynamics. A line of work focused on systems with dynamics and removed the requirement for a steering function by starting from first principles to achieve AO properties (Li et al 2016; Littlefield and Bekris 2018).
The analysis starts with a Monte Carlo search tree, which performs random selection of nodes and random propagation of controls to expand in the underlying state space and does not depend on a steering function. Such a Monte Carlo approach is shown to be AO, but also not a practical solution due to slow convergence rate. A BestNear variant of the basic Monte Carlo tree, however, was shown to be both asymptotically (near)optimal and have a practical convergence rate. The variant prioritizes the selection of nodes with good path quality within a neighborhood of the random sample, but still uses Monte Carlo propagation. It can achieve strict AO properties by reducing the neighborhood size for this selection choice over computation time giving rise to an anytime behavior. The variant can be further improved computationally by sparsifying the underlying tree data structure and storing only nodes that locally have good path quality, giving rise to the SST algorithm (Fig 4). The use of heuristics in this context (Littlefield and Bekris 2018) has shown to further speed up practical performance. This model of analyzing treebased planners has been extended to study in general the properties of RRTbased planners in both kinematic and kinodynamic domains (Kleinbort et al 2019b).
1 ;
2 if then ;
3 for do
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5
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AOA: Another direction for achieving desirable properties is a metaalgorithm, referred to as and shown in Algo 3 (Hauser and Zhou 2016). The approach provides AO properties as long as the algorithm is probabilistically complete (PC) in the statecost space and also achieves exponential convergence. An algorithm is PC in the statecost space if it is guaranteed to find a solution within a desired costbound, if one exists, given enough time. The algorithm inspects the problem of optimality in the statecost space, and repeatedly calls the algorithm to produce a solution within the best cost bound discovered so far. Note that algorithm in Algo 3 takes the costbound as an argument and returns the first solution discovered within the input cost bound. By reducing the cost bound, the solution converges to the optimal cost, and the PC and exponential convergence properties of ensure that the expected running time is also finite. Recently, specific attention (Kleinbort et al 2019a) has been paid to designing an AO algorithm based on RRT that explores the statecost space.
NearOptimal RGG: Recent work revisits the theoretical underpinnings of random geometric graphs, inspecting the properties of roadmapbased methods in motion planning under a simpler analysis model that uses a hypercube construction in a unit hypercubic configuration space Solovey et al (2018). The insights from previous work allow to deduce that PRM using a connection radius of
is probabilistically complete, and imposes a bounded stretch factor degradation over the optimal path (nearoptimality).
Bridging Theoretical Guarantees and Practical Performance
Asymptotic optimality provided theorists with a lens to inspect algorithmic models that allow path quality guarantees. AO comes at the price of computation overhead per iteration when compared to simply ensuring probabilistic completeness. In practice, there is a significant gulf between finding an initial solution and finding the optimal solution. It is not clear how the asymptotic worst case analysis models for PC and AO properties translate into practical performance, and guarantees at finite time. Though asymptotic optimality is an evidently theoretical construct, it is important that AO motion planning algorithms exhibit in practice anytime properties, i.e., they report good quality initial solutions fast and continue improving path quality as a function of computation time.
Relaxed but Practical Guarantees: Computational tradeoffs can be made by foregoing on the strictness of AO properties for asymptotic nearoptimality(Marble and Bekris 2013).
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These methods are based on underlying graph spanners (Algo 4), which refer to subgraphs that tradeoff connectivity and pathcosts. This tradeoff is denoted by the parameter which bounds the path degradation over between the pairwise nodes during the addition of edges in roadmap construction. The IRS algorithm crucially differs from an incremental variant of PRM in the highlighted steps that perform the trelaxation during edge additions. These methods result in sparser data structures (Dobson and Bekris 2014) that offer benefits of computational and storage savings. Sparser variants Li et al (2016) of treebased planners prune nodes that reach similar parts of with a worse path. The sparsity of the data structures can be seen in Fig 6 and demonstrates the practical power of these relaxations.
The probabilistic asymptotic guarantees occur after infinite samples, but it is important to have a model for deciding on the number of samples a motion planner should be executed in practice. These are studied as finite time properties of AO methods (Dobson and Bekris 2013), which have yielded insights into the tradeoffs of computation and solution quality expected from these algorithms.
There have been other models for studying nearoptimality for treebased approaches (Salzman and Halperin 2016) and random geometric graph based ones (Solovey et al 2018).
Improving Computational Efficiency: There are a variety of ways to improve the computational efficiency of the AO planners, including through parallelizing (Bialkowski et al 2011) and caching collision checking information (Bialkowski et al 2013). A way to minimize collision checking is to perform it lazily (Haghtalab et al 2018). Sampling strategies that focus on regions of the that can improve existing solutions have been applied to both singleprocessor planners (Gammell et al 2015) and in parallel search processes with shared information (Otte and Correll 2013). Unlike RRT which uses local rewiring, global cost information propagation (Arslan and Tsiotras 2013) allows faster convergence in singleshot and replanning frameworks (Otte and Frazzoli 2016). Heuristics have also been incorporated into elements of kinodynamic planning (Littlefield and Bekris 2018). These optimizations can let AO algorithms quickly discover highquality solutions, while improving on these solutions over time. Such methods are called anytime algorithms. Guidance can also arise out of human demonstrations which the planner can learn in order to guide an underlying AO search strategy Bowen and Alterovitz (2016). On realworld systems, computation limits can be sidestepped by relying on the cloud (Ichnowski and Alterovitz 2016; Bekris et al 2015). Hybrid approaches (Luna et al 2013; Choudhury et al 2016) have combined optimization strategies to refine the solutions obtained in an AO framework.
AO Planners for Extensions of the Basic Problem
The progress on AO properties has allowed extending such guarantees to new problem domains. In particular, most of the above algorithms are applicable to kinematic domains given a Euclidean norm as an optimization objective. Below is a list of effort that extend AO algorithms and analysis models to interesting robotics problems where the assumptions of the original methods may no longer apply.
Kinodynamic Planning: The kinodynamic motion planning problem deals with motion planning for a robotic system with significant dynamics, i.e., the planning has to consider and account for velocities, accelerations (and other higher order dynamics). The approaches based on random geometric graphs assume the existence of a steering function to guarantee that two nearby configurations can always be connected. In a kinodynamic problem, this does not always exist. An AO approach was proposed for the specifc case of systems with linearized dynamics (Webb and Van Den Berg 2013).
A general framework focused on a novel analysis specifically for kinodynamic domains without access to a steering function (Li et al 2016). The underlying treebased approach performs an efficient forward search that leverages Monte Carlo propagation, i.e., random controls executed for random duration, to ensure asymptotic coverage of the space. While such a random behavior is desirable for the analysis, practical performance can be improved using approaches that employ heuristics efficiently (Littlefield and Bekris 2018).
A variety of PC algorithms, including RRT, have been known to deal with kinodynamic problems. The AO framework (Hauser and Zhou 2016) demonstrated that an underlying PC approach can be repeatedly restarted to yield asymptotic optimality, under certain conditions. This allows an elegant solution to AO kinodynamic planning as well.
Motion Planning with Constraints: Many motion planning problems introduce constraints that force feasible solutions to lie on lower dimensional manifolds of . Consider the difference between planning to move a robotic arm versus moving it while keeping a grasped object upright. Such manifolds have 0volume in the fully dimensional . This proves problematic for any samplingbased strategy when the lowerdimensional manifold cannot be explicitly defined.
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Approaches aim to ensure that samples and connections are discovered in such domains, while preserving theoretical properties (Kingston et al 2018).
Some variants use projection operators (Berenson et al 2011) to reach the constraint manifold, while others operate on the tangentspace of the constraint manifold by decomposing local neighborhoods into atlases (Jaillet and Porta 2013). A way to look at the problem is to decouple constraint satisfaction from the underlying motion planner (Kingston et al 2017). The treatment of such constraints is also important in the context of integrated task and motion planning (VegaBrown and Roy 2016).
Multirobot Motion Planning: The challenge posed by planning for multiple robotic systems is the explosion in the size of the configuration space. Centralized multirobot planning methods (Solovey et al 2015) have been argued to be AO (Shome et al 2019) under a samplingbased scheme that builds a graph in the configuration space of each robot and performs an online search over an implicit representation of a tensorroadmap in the configuration space of all the robots (shown in Fig 7). The key idea is that the configuration space of robots can be seen as the Cartesian product of the constituent spaces . Accordingly, configurations and nodes can be split into the respective components in the constituent spaces. Algo 5 demonstrates the construction of AO roadmaps (ao_rm) in each such space. The combination of these roadmaps has been shown to be AO in the entire and is called the tensorroadmap
Instead of explicitly creating and storing , the idea is to implicitly search it online through a tree that is constructed over the nodes in . Avoiding storage and enumeration of the tensorroadmap has important memory and computation benefits. The highlighted parts of Algo 5 show the differences between the highlevel description of dRRT and RRT in the construction of an AO search tree over . The extension to a new node is performed by exploring adjacent vertices in each constituent roadmap . Both the selection and extension is heuristically biased using solutions for the single robot on their corresponding roadmaps. As and grows to infinity, the algorithm is guaranteed to converge to the optimal solution that is contained in .
Beliefspace Planning: Most realworld applications of robotics have to contend with uncertainty. There are efforts (Aghamohammadi et al 2012; Chaudhari et al 2013), which have focused on extending properties of samplingbased planning algorithms to the domain of beliefspace planning where instead of planning over configurations, the method has to reason about a distribution of such states that correspond to its belief. Recent work (Littlefield et al 2015) has also demonstrated the considerations and conditions under which such problems can be solved using AO algorithms that do not rely on steering functions.
Properties  Algorithms 

AO kinematic  PRM, RRT, RRG (Karaman and Frazzoli 2011), 
RRT (Arslan and Tsiotras 2013), FMT (Janson et al 2015),  
BIT (Gammell et al 2015), RRT (Otte and Frazzoli 2016)  
AnO kinematic  IRS (Marble and Bekris 2013), SPARS (Dobson and Bekris 2014), LBTRRT (Salzman and Halperin 2016) 
kinodynamic  SST (Li et al 2016), AO (Hauser and Zhou 2016), 
DIRT (Littlefield and Bekris 2018),  
RRT2.0 (Kleinbort et al 2019a)  
constraints  atlasRRT (Jaillet and Porta 2013) 
multirobot  dRRT (Shome et al 2019) 
5 Examples of Application
With the emergence of AO algorithms in motion planning, there has been a push from researchers to ensure that methods in applications afford such guarantees. In practice this means that as increased computational resources become available, the methods are able to take advantage of them and return higher quality solutions.
Some example applications (Fig 8) where AO motion planning techniques have been used include the following:

Selfdriving cars (hwan Jeon et al 2013): AO motion planners has been applied to highspeed driving applications on car models with dymamics and fully integrated autonomy systems on shared roadways.

Space robotics (Littlefield et al 2019): Space exploration is a pertinent application of robots, which involves deployment in highly unstructured environments. New emerging rovers, such as the hybrid softrigid mechanisms based on the principles of tensegrity, introduce unique planning difficulties, which have been approached with AO motion planners.

Medical robotics (Patil et al 2015): Robustness and quality of solutions is important in medical applications and promote AO considerations in the utilized planners. Medical robots need to carefully maneuver surgical or diagnostics tools through the human body safely and effectively.

Robot design (Baykal et al 2019): The design process for a robot can be seen as a model that, if optimized for a specific application, can significantly help in addressing challenging problems.
6 Future Direction for Research
Future research can potentially help into bridging the gap between theoretical properties and practical performance. For instance, the typical setup for the analysis of samplingbased planners accounts for worstcase reasoning. There is the potential for studying the expected behavior of these methods in typical instances and further clarifying the conditions under which samplingbased planners face issues in converging to highquality paths. The inspection of convergence rates as well as efficient data structures can provide insights regarding the practical and predictable deployment of AO methods to different domains.
Computing Platforms: There has been a recent push to evolve the algorithms to leverage modern computing hardware. Parallelization (Bialkowski et al 2011) has been a focus in different aspects of AO motion planning. Custom designed chipsets (Murray et al 2016) can optimize specific motion planning queries and lead to orders of magnitude speedups. Moreover, robots of the future will potentially be part of a collective cloud infrastructure, which can share information among motion planning methods (Bekris et al 2015; Ichnowski and Alterovitz 2016). GPUs, which have led to a revolution in machine learning applications, can also be used to improve the efficiency of motion planners (Ichter et al 2017), given efficient parallel primitives (Pan and Manocha 2012).
Learning: Learningbased approaches (Faust et al 2018) are being integrated with samplingbased planners so to speedup and improve the performance of key planning components. Learning how to perform sampling and identifying the key latent spaces for complex robotic system (Ichter and Pavone 2019) provides an intriguing avenue for augmenting the capabilities of algorithmic motion planning.
Planning under Uncertainty: Further analysis is needed in order to be able to argue AO for motion planners in this domains (Littlefield et al 2015) as well as significant progress towards achieving computational efficiency.
Integrated Task and Motion Planning: Many applications, such as manipulation, require identifying the sequence of motion planning challenges that need to be addressed for solving a task together with solving each of these motion planning instances. Recent progress in such integrated task and motion planning has set the foundations for understanding the conditions for asymptotic optimality in this domain (VegaBrown and Roy 2016), while it provides opportunities for applying AO planners in important domains, such as object rearrangement (Shome et al 2018).
Footnotes
 email: kostas.bekris@cs.rutgers.edu
 email: rahul.shome@cs.rutgers.edu
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