Robot Motion Planning Using Adaptive Hybrid Sampling in Probabilistic Roadmaps

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1 electronics Article Robot Motion Planning Using Adaptive Hybrid Sampling in Probabilistic Roadmaps Ashwin Kannan, Prashant Gupta, Rishabh Tiwari, Shubham Prasad, Apurv Khatri and Rahul Kala * Department of Inmation Technology, Indian Institute of Inmation Technology, Allahabad, Deoghat, Jhalwa, Allahabad , India; iit @iiita.ac.in (A.K.); iit @iiita.ac.in (P.G.); iit @iiita.ac.in (R.T.); iit @iiita.ac.in (S.P.); iit @iiita.ac.in (A.K.) * Correspondence: rkala001@gmail.com; Tel.: Academic Editor: Mostafa Bassiouni Received: 26 December 2015; Accepted: 21 March 2016; Published: 6 April 2016 Abstract: Motion planning deals with finding a collision-free trajectory a robot from current position to desired goal. For a high-dimensional space, sampling-based algorithms are widely used. Different sampling algorithms are used in different environments depending on nature of scenario and requirements of problem. Here, we deal with problems involving narrow corridors, i.e., in order to reach its destination robot needs to pass through a region with a small free space. Common samplers used in Probabilistic Roadmap are unim-based sampler, obstacle-based sampler, maximum clearance-based sampler, and Gaussian-based sampler. The individual samplers have ir own advantages and disadvantages; ree, in this paper, we propose to create a hybrid sampler that uses a combination of sampling techniques motion planning. First, contribution of each sampling technique is deterministically varied to create time efficient roadmaps. However, this approach has a limitation: The sampling strategy cannot adapt as per changing configuration spaces. To overcome this limitation, sampling strategy is extended by making contribution of each sampler adaptive, i.e., sampling ratios are determined on basis of nature of environment. In this paper, we show that resultant sampling strategy is better than commonly used sampling strategies in Probabilistic Roadmap approach. Keywords: motion planning; sampling-based motion planning; configuration space; probabilistic roadmap; robotics 1. Introduction The interest in making machines smart and giving m ability to act autonomously in everyday situations has resulted in an increased interest in robotics. Modern day robots are increasingly becoming sophisticated to serve ir human masters with very little specifications. Robot Motion Planning [1,2] is a widely studied problem in robotics, which aims to make a robot reach a desired location or goal from source. The robot here may be a mobile robot, an aerial robot, a robotic hand or a manipulator, a humanoid robot, or any or kind of robot. The path of robot computed by motion planning algorithm must be collision-free, must be of shortest length possible, must maintain a respectable clearance from obstacles around, must be smooth, and must be computable in small computation times. The configuration of robot is a specification of all parameters of robot determining its pose as it travels from source to goal. The configuration space (C) is collection of all possible robot configurations. A configuration (q) is called collision-free if robot placed at configuration q does not collide with any obstacle or with itself. A collection of all collision-free configurations constitutes free configuration space (C free ). Correspondingly, obstacle-prone configuration Electronics 2016, 5, 16; doi: /electronics

2 Electronics 2016, 5, 16 2 of 13 space is a collection of all configurations where a collision happens, and is compliment of free configuration space over C (i.e., C obs = C/C free ). A path is thus function τ: [0, 1] Ñ C free, starting from τ(0) = q source to τ(1) = q goal, such that every intermediate point τ(s) is collision-free (i.e., τ(s) ɛ C free, 0 ď s ď 1). Here, q source is source configuration while q goal is goal configuration. A popular technique in motion planning is to make free configuration space discrete by packing it in grids [3]. Each grid is assumed to be connected to some neighboring grids if straight line connection between grids is collision-free. The problem is hence modeled as a graph search problem, over which graph search algorithms like A* algorithm [4] can be applied. Similarly, geometric algorithms exploit known geometry of obstacle to compute path. In case of low dimensional problems, grid-based algorithms and geometric algorithms are widely used; however, higher dimensional problems, exact motion planning is NP hard, so sampling-based algorithms are more common. The sampling-based algorithms sample out configuration space in pursuit of computing a path between source and goal. A common sampling-based algorithm is Probabilistic Roadmap (PRM) [5]. PRM is a multi-query algorithm that has two phases: a construction phase and a query phase. The construction phase is offline and generates a roadmap (or a graph). The vertices of roadmap are randomly generated samples which belong to free configuration space (C free ). Every sampled configuration (q) is checked connectivity typically with k closest neighboring samples (q 2 ). Anor common strategy checks connectivity with all samples that lie at a distance of less than k units. In order to check connectivity of q and q 2 by an edge, a local planning algorithm is used. A common local planning algorithm checks a straight line connectivity between samples and adds an edge if straight line connection between q and q 2 is collision-free. In query phase, source (q source ) and goal (q goal ) are temporarily added to roadmap and possibility of an edge is checked with neighboring samples already in roadmap. The resultant roadmap is a graph with source and goal; thus, any graph search algorithm may be used to compute path. PRM is a probabilistically complete algorithm, meaning that probability of finding a solution, if one exists, tends towards 1 as number of samples (or computation time in roadmap construction) tend to infinity. In worst case, roadmap generation will cover entire configuration space by placing samples, meaning that all configurations in path will be sampled out, at which stage roadmap will be dense enough to return a path. Practically, PRM returns a solution in small computation times, even problems with a very high dimensionality. Since a shortest path graph search algorithm is used to compute a path, objective of a small path length is naturally represented in approach, given that enough samples are used to represent roadmap. However, one of limitations of PRM is that paths are very steep at points of turn. A non-holonomic robot will not be able to follow se paths, as y do not obey non-holonomic constrains. Smoothing is done after computing a path in order to make paths controllable by a non-holonomic robot. Smoothing must make sure that it does not make feasible paths as infeasible, which reason it is suggested to add enough clearance in path and use small local optimizations. The permance of PRM largely depends upon sampling technique used [6,7]. A variety of sampling techniques have been proposed in literature, each suited a certain scenario. In this paper, we propose using all sampling strategies in a hybrid architecture to create a planner that is able to work in different types of mixed environments. The optimal contribution of different strategies may vary with time; hence, a deterministic rule is used to change contributions so as to imitate optimal growth of every sampling strategy along with time or changing face of roadmap. Furr, aim is to make a sampling technique that is able to handle different types of scenarios. The deterministic rule does not model different scenarios. Hence, approach is made adaptive so as to assess operational scenario and hence fix contributions of different sampling techniques used. The resultant strategy is experimentally found to be better than unim sampling.

3 Electronics 2016, 5, 16 3 of 13 Numerous hybridization schemes have already been used in literature. Hsu et al. [6] adaptively hybridized different samplers based on ir usefulness, where usefulness is indicated by ability to add new disconnected samples or connecting two disconnected components of roadmap. Theree, adaptation largely holds in cases of complex narrow corridors. The aim of proposed work is also to adapt samplers when multiple paths to goal have been found and re are no disconnected components in roadmap, wherein such measures do not hold. Similarly, Kurniawati et al. [8] hybridized samplers, also proposed sampling as per connectivity of workspace, which an adjacency graph was constructed. Assumption of workspace geometry is a particularly strong assumption to make in motion planning, which can also mean long computation time to initially interpret workspace geometry to produce corresponding graph complex scenarios with multiple obstacles. Rodrıguez et al. [7] suggested classifying different types of regions based on obstacle density, and to correspondingly select samplers. The hybridization scheme is hence space-based, unlike proposed approach, wherein hybridization scheme is time-based. This ms a fundamental difference in approach. The adaptation is as per motivation, wherein our intention is to create a sampling technique which gives best roadmap any general situation. Hence, time cannot be invested heavily at start to select and classify different regions, while one might, at least in simple scenarios, want a path to be quickly computed in same duration. Similarly, Morales et al. [9] furr used edge connectivity measures to decide region difficulty. Our future research aims are also targeted at sampling techniques that boost samples in small regions around narrow corridors, which superficially increases connectivity of samples such that, when hybridizing with such custom samplers, connectivity measures cannot be used. A popular method single query problems using a sampling-based approach is Rapidly-exploring Random Trees (RRT) [10]. The method grows a tree from source towards goal in configuration space and stops when a path is found. RRT* [11] algorithm continues searching in pursuit of acquiring optimal paths. The Rapidly-exploring Random Graph [12,13] algorithm extends concept by use of a graph that grows with time. It is also common to maintain multiple trees [14,15] instead of just one, and to integrate outputs. Hybridization has also been applied on RRTs. As an example, Denny et al. [16] proposed using visibility, which was approximately computed as a criterion to select technique used to grow RRT. Similarly, Jouandeau and Cherif [17] solved problem of narrow corridors in RRT by using a modified Gaussian sampling. A lot of research has taken place in use of sampling-based approaches time efficiency and optimal motion planning of robots. Attempts have been made to make generated roadmap homotopic conscious in order to quickly generate all homotopic groups of paths [18]. The approach first adds a set of vertices and edges, and subsequently adds more vertices and edges so as to induce useful cycles in roadmap that ultimately lead to discovery of paths in different homotopic groups. Similarly, an approach uses RRT as local search algorithm while using PRM [19]. RRT is able to find non-straight line paths between vertices and thus results in enhanced connectivity between vertices at a small additional computation time, which severely improves metrics of completeness and optimality. Spark PRM [20] has been proposed to better discover narrow corridors by making RRT expansion from inside narrow corridors. It is still easy to find a few samples inside narrow corridors by using narrow corridor centric sampling techniques. The problem is usually connection of se samples to samples outside, constituting most of roadmap. A traversal from inside to outside of narrow corridor caters to such connections. Lazy PRM [21] attempts to reduce computation time in roadmap generation by delaying collision checking until search is made in query phase. The approach is hence suggested when using PRM a single query, in which case algorithm perms faster by eliminating collision-checks in regions that are potentially not in optimal path. When using multiple queries, all edges are eventually checked collisions. So far, little research has been done in effective use of sampling

4 Electronics 2016, 5, 16 4 of 13 of PRM generation of roadmaps that suit a variety of scenarios of operation. This paper is a step in same direction by proposing a hybrid and adaptive sampling technique PRM. This paper is organized as follows. Section 2 discusses different sampling techniques. Section 3 motivates and designs hybrid sampling technique, presenting also a deterministic rule to vary contributions of individual techniques. Section 4 carries ward discussions by creating an adaptive sampler problem. Section 5 presents experimental results, while Section 6 provides concluding remarks. 2. Sampling Strategies The strength of PRM is largely attributed to sampling strategy used. A sampling strategy determines how samples are generated in free-configuration space (C free ) and become vertices of roadmap. A common sampling strategy used in PRM is unim sampling strategy, wherein samples are unimly generated in free configuration space. This sampling-based approach faces a severe problem when path goes through a narrow corridor. A narrow corridor is a very small volume of free configuration space, surrounded by obstacle prone configuration space. As volume of free space through a region of path is small, probability of a random sample being generated inside narrow corridor is small. This means a very high number of samples need to be generated to discover narrow corridor, which increases computation time in both offline construction phase and online query phase. The optimal paths see robots making turns around obstacles, while traveling in a straight line away from obstacles. Hence, it is preferred to keep samples near obstacles. For this reason, a better sampling strategy is used called as obstacle-based valid sampling strategy [22]. This strategy generates samples near obstacle boundary. Since samples have a limited connectivity to only a few neighboring samples, it cannot be ascertained that optimal path can be computed using this strategy alone. The generation of samples is done by taking a sample in collision-prone configuration space (q obs ) and n anor sample in collision-free configuration space (q free ). A sample at a proportionate distance of λ from q obs towards q free (similarly 1 λ from q free towards q obs ) is given by λ q free + (1 λ) q obs. By increasing value of λ, sample travels from q obs towards q free. The sample is moved until a transition from obstacle to non-obstacle configuration space is recorded, which is when sample crosses obstacle boundary and lies at C free. The sample in C free is accepted as new sample. The process is explained by Algorithm 1. Algorithm 1: Obstacle-Based Valid Sampling Input: C, C obs, C free Output: Valid Sample q while true q obs Ð random(c) if q obs ɛ C obs, break end while while true q free Ð random(c) if q free ɛ C free, break end while q Ð q obs, λ Ð 0 while q ɛ C obs q Ð λ q free + (1 λ) q obs increment λ by a small step end while return q

5 Electronics 2016, 5, 16 5 of 13 Anor similar sampling technique is Gaussian valid configuration sampler [23], which also tries to generate samples near obstacle boundary. The technique attempts to directly sample at obstacle boundary, rar than generating a sample in obstacles and n making same reach boundary. The traveling time of sample is hence saved. The technique generates a sample in obstacle prone configuration space (q obs ) and a second sample at a normal distribution around q obs. The direction of second sample is taken randomly. Let N(0, σ) be normal distribution with mean 0 and variance σ. The variance σ is an algorithm parameter set to be a small proportion of configuration space. If generated sample is in free-configuration space, sample is accepted. Even though samples are generated directly at obstacle boundary, failure ratio of technique can be very high, in which case technique re-tries generation of samples. The algorithm is explained in Algorithm 2. Algorithm 2: Gaussian-Based Valid Sampling Input: C, C obs, C free Output: Valid Sample q while true while true q obs Ð random(c) if q obs ɛ C obs, break end while q Ð q obs + N(0, σ) if q ɛ C free, break end while return q The above strategies generate samples near obstacle boundary and ree can result in a very small clearance of resultant path. The maximum clearance-based valid sampler attempts to generate samples which have a high clearance. Samples with high clearance can also aid in better connectivity with neighboring samples. The sampler is given by Algorithm 3. The algorithm attempts to increase clearance a total of K attempts. Here, K is a small constant like 10. Algorithm 3: Maximum Clearance-Based Valid Sampling Input: C, C obs, C free, K Output: Valid Sample q maxclearance Ð 0 qmaxclearance Ð NULL i Ð 0 while i < K q Ð random(c) if q ɛ C free if Clearance(q) > maxclearance maxclearance = Clearance(q) qmaxclearance Ð q end if end if i Ð i + 1 end return qmaxclearance

6 Electronics 2016, 5, 16 6 of Deterministic Hybrid Sampling The different samplers discussed in Section 2 have a varying permance based on scenario of operation. For example, unim-based sampler perms better in obstacle-less regions whereas obstacle-based sampler perms better in narrow corridors and scenarios of high obstacle density. The individual samplers have ir own advantages and disadvantages depending upon scenario and requirements. Hence, proposal is to use a hybrid sampling technique, wherein all se sampling algorithms are used in different contributions. A hybrid of se samplers could significantly improve permance of PRM algorithm. The segments of configuration space suited a specific sampler can be probabilistically catered to by same sampler. The hybrid sampler specifies a selection probability every sampler. The samplers considered are same as discussed in Section 2. Every time a sample needs to be generated, one of samplers is selected and same is used generating a random sample used generation of roadmap. Hence, overall roadmap witnesses samples generated by each of discussed samplers. While it appears that hybrid sampling is capable of adding advantages of samplers, while eliminating limitations, resultant hybrid strategy has a big limitation that permance largely depends upon selection probability of individual samplers, which is an algorithm parameter and needs to be fixed by trial and error. PRM, being an anytime algorithm whose execution can be stopped anytime to acquire a solution while solution quality improves on increasing execution time, sees a transition in growth of roadmap from sparse to dense. The selection probability largely depends upon stage of roadmap generation; thus, selection probabilities cannot be hard fixed. Hence, we introduce a deterministically varying hybrid sampler. Here, selection probabilities of individual samplers is not fixed but is allowed to vary using a pre-defined rule depending upon time. The rule to vary selection probabilities is simple to specify. The idea behind rule is that it is useful to sample in difficult regions ( regions with higher number of obstacles and narrow corridors) at first, but number of such samples decrease with time. Initially, roadmap is sparse, and it is important to sample out narrow corridors and regions around obstacles to find at least some path every source and goal query combination. Later, as roadmap density increases, nearly all obstacles and narrow corridors are already out. Theree, we gradually shift our focus from sampling in difficult regions to sampling unimly over entire configuration space. This aids in acquiring redundant cycles, thus contributing to optimality, as well as aiding in connectivity between distant samples. We assign an initial selection probability to each sampler and change se selection probabilities along with time. The obstacle-based sampler and Gaussian-based sampler are given high initial selection probability so as to sample difficult regions early on, whereas unim-based sampler has a high terminal selection probability, indicating shift in our sampling strategy. All selection probabilities at all times should add up to one, and accordingly normalization is done. Let selection probabilities of different samplers at time t be denoted as follows: P O (t) obstacle-based sampler, P G (t) Gaussian-based sampler, P M (t) maximum clearance-based sampler, and P U (t) unim-based sampler, such that P O (t) + P G (t) + P M (t) + P U (t) = 1. The initial selection values P O (0), P G (0), P M (0), and P U (0) are fixed and are algorithm parameters. Furr, it is assumed that selection probabilities converge after a timeout T, after which ir values are not changed with time but remain constant. The convergent selection probabilities P O (T), P G (T), P M (T), and P U (T) are also algorithm parameters. It should be obvious that final selection probability of unim-based sampler will be higher than obstacle- or Gaussian-based sampler. The algorithm a hybrid sampler, varying with time is given in Algorithm 4.

7 Electronics 2016, 5, 16 7 of 13 Algorithm 4: Deterministically Varying Hybrid Sampling Input: C, C obs, C free, Time t Output: Valid Sample q Calculate P O (t), P G (t), P M (t) and P U (t) from Equation (1) Normalize P O (t), P G (t), P M (t) and P T (t) r Ð random number between 0 and 1 if r < P O (t), return Obstacle based Valid Sampler (Algorithm 1) else if r < P O (t) + P G (t), return Gaussian based Valid Sampler (Algorithm 2) else if r < P O (t) + P G (t) + P M (t), return Maximum Clearance based Valid Sampler (Algorithm 3) else return Unim Valid Sampler For simplicity, change in selection probabilities with time is modeled as a linear function of time, from initial value at time 0 to final value at time T, while selection probabilities remain constant after time T. The selection probabilities are given by Equation (1). $ & P X p0q ` pp X ptq P X p0qq t P X ptq T % P X ptq if t ď P to, G, M, Uu (1) if t ą T Taking a larger timeout will ensure a gradual shift from obstacle-based sampling to a unim sampling. Since obstacle-based sampler returns a point in very vicinity of obstacle, it will also help in curve smoothing part of motion planning while keeping path length small. The sampling technique is given by Algorithm Adaptive Hybrid Sampling The problem with deterministically varying hybrid sampler discussed in Section 3 is that it does not capture change in operating scenario. In robot motion planning, scenarios can vary from ones densely occupied by obstacles to ones with a very small obstacle density; some of se may not have any narrow corridors, while some ors may have numerous elongated narrow corridors. A specific initial and final selection probability cannot suit all types of scenarios. Hence, intention is to make se parameters adaptive. Our next algorithm, adaptive sampler is a generalization of algorithm of Section 3 and makes it independent of scenario under which robot is operating. Obstacle density is chosen as metric of denoting type of scenario. The factor is relatively easy to calculate in small computation times and easily represents difficulty of scenario. Correspondingly, initial and final selection probability of individual samplers is set as a function of obstacle density. This makes sampler adaptive in nature, as it can adjust itself depending on scene. The first step in this algorithm is to calculate approximate obstacle density (ρ) of scenario. This is done by taking a number of random samples (Q) from environment and calculating number of invalid samples among m. The density (ρ) is given by Equation (2). ρ q P Q X C obs Q (2) The density n needs to be used to set initial and final selection probabilities of samplers. The same is done using a simple rule. If density is high, n we should prefer a higher initial selection probability of obstacle-based sampler. An adaptive algorithm based on se ideas is given in Algorithm 5. If density is low, n we should prefer a higher initial selection probability of unim-based sampler. All probabilities add up to one.

8 Electronics 2016, 5, 16 8 of 13 Algorithm 5: Adaptive Hybrid Sampling Input: C, C obs, C free, Time t Output: Valid Sample q if t = 0 Q Ð N random samples Calculate ρ from Equation (2) Calculate P O (0), P G (0), P M (0) and P U (0) using Equation (3) Calculate P O (T), P G (T), P M (T) and P U (T) using Equation (4) end if Calculate P O (t), P G (t), P M (t) and P U (t) from Equation (1) Normalize P O (t), P G (t), P M (t) and P T (t) r Ð random number between 0 and 1 if r < P O (t), return Obstacle based Valid Sampler (Algorithm 1) else if r < P O (t) + P G (t), return Gaussian based Valid Sampler (Algorithm 2) else if r < P O (t) + P G (t) + P M (t), return Maximum Clearance based Valid Sampler (Algorithm 3) else return Unim Valid Sampler The initial selection probability of different samplers is hence given by Equation (3). P O p0q α O ρ, P G p0q α G ρ, P M p0q α M, P U p0q 1 pp O p0q ` P G p0q ` P M p0qq (3) Here, α O, α G, and α M are algorithm constants. Care is given in setting se constants such that all probabilities lie between 0 and 1, and all add up to 1. Normalization may hence be required. Similarly, final selection probabilities of samplers are given by Equation (4). P O ptq β O ρ, P G ptq β G ρ, P M ptq β M, P U ptq 1 pp O ptq ` P G ptq ` P M ptqq (4) Similarly, β O, β G, and β M are similar algorithm constants. The selection probability at any time can be linearly interpolated using Equation (1). The resultant algorithm is given by Algorithm Results The experiments are done using Open Motion Planning Library (OMPL) [24]. The OMPL has a framework sampling-based motion planning. Experiments are done both on OMPL App, which is a standalone application using OMPL and MoveIt [25], a wrapper around OMPL mobile manipulation in Robot Operating System (ROS) environment. To test hybrid and adaptive strategies, we ran algorithm on articulated robots in three-dimensional environments. Experiments were done using benchmarks available in ROS environment and in OMPL App. Due to space constraints, only two scenarios are discussed here. The first scenario is called as Twistycool and features a toy robot in SE(3) configuration space that has to surpass a narrow corridor region. The scenario is shown in Figure 1a. The second scenario is Alpha-1.5 scenario in which a tangled robot has to untangle itself with a similar looking environment. The configuration space is again SE(3). The scenario is shown in Figure 1b. Both scenarios have a narrow corridor which makes m very difficult to handle. The parameters of algorithm were fixed by trial and error. The algorithms were executed different parameter settings. Settings which seemed to perm reasonably well in a variety of scenarios were finally accepted. The values of parameters are mentioned in Table 1. The value of timeout (T) is kept as 100 throughout.

9 tangled robot has to untangle itself with a similar looking environment. The configuration space is again SE(3). The scenario is shown in Figure 1b. Both scenarios have a narrow corridor which makes m very difficult to handle. The parameters of algorithm were fixed by trial and error. The algorithms were executed different parameter settings. Settings which seemed to perm Electronics reasonably 2016, well 5, 16 in a variety of scenarios were finally accepted. The values of parameters 9 ofare 13 mentioned in Table 1. The value of timeout (T) is kept as 100 throughout. Figure 1. Test Scenarios Twistycool Problem; Alpha-1.5 Problem. Figure 1. Test Scenarios Twistycool Problem; Alpha-1.5 Problem. Table 1. Parameter Settings. Table 1. Parameter Settings. Initial Selection Final Selection Initial Selection Final Selection Serial Sampler Initial Probability Selection Final Probability Selection Initial Probability Selection Final Probability Selection Serial No. Sampler (Deterministic) Probability (Deterministic) Probability Probability (Adaptive) Probability No. (Adaptive) (Deterministic) (Deterministic) (Adaptive) (Adaptive) 1 Obstacle-Based Sampler ρ 0.1 ρ 12 Gaussian-Based Obstacle-Based Sampler ρ 0.1 ρ 2 Maximum Gaussian-Based Clearance-Based Sampler ρ 0.1 ρ 3 Maximum Clearance-Based Sampler Sampler 4 Unim-Based Sampler (above) 1 (above) Electronics , 5, Unim-Based 16 Sampler (above) 1 (above) 9 of 12 The simulations were permed using benchmarking libraries provided by OMPL. We first tested our planner on an articulated robot in a three-dimensional environment containing a narrow corridor over Twistycool benchmark. A total of 20 runs were taken each algorithm, and time limit was restricted to 20 s per run. The source and destination configurations robot were taken as specified in benchmarking test suite. In first scenario, default PRM planner gives exact solution in 30% of total runs, whereas adaptive planner gives exact solution in 65% of of total total runs, runs, and and deterministically varying varying planner planner solves solves 35% of 35% of total test-cases. total test-cases. Since only Since few only cases few resulted cases resulted in a valid in a trajectory, valid trajectory, path length path metric length ismetric not a reliable is not a metric reliable asmetric per our as objectives. per our objectives. The results The aresults shown are inshown Figurein 2. Figure 2. Figure 2. Results Twistycool Problem Success Rate; Solution Length. Figure 2. Results Twistycool Problem Success Rate; Solution Length. The second scenario is Alpha-1.5 problem. This scenario is more complex and ree it is chosen rigorous testing and parameter tuning. First, comparisons are made with base samplers used approach. The approach exceeding all base samplers is an indication that a hybrid of samplers exceeds permance of all individual samplers and does not simply average out results. In this scenario, unim-based sampler gives exact solution in only 10% of

10 Figure 2. Results Twistycool Problem Success Rate; Solution Length. Electronics 2016, 5, of 13 The second scenario is Alpha-1.5 problem. This scenario is more complex and ree it is chosen Therigorous second scenario testing is and Alpha-1.5 parameter problem. tuning. ThisFirst, scenario comparisons more complex are made and ree with it is base samplers chosen used testing approach. and parameter The approach tuning. exceeding First, comparisons all base are samplers made with is an base indication samplers that a used approach. The approach exceeding all base samplers is an indication that a hybrid of hybrid of samplers exceeds permance of all individual samplers and does not simply average samplers exceeds permance of all individual samplers and does not simply average out out results. In this scenario, unim-based sampler gives exact solution in only 10% of results. In this scenario, unim-based sampler gives exact solution in only 10% of total total runs, runs, obstacle-based obstacle-based sampler sampler gives a gives solution a solution in about in 3% about of 3% total of runs, whereas total runs, adaptive whereas adaptive planner planner gives gives exact solution exact in solution 27% of in total 27% runs, of showcasing total runs, a significant showcasing improvement a significant in improvement permance. in The permance. deterministically The deterministically varying planner solves varying 17% planner of total solves test-cases. 17% of The total path testcases. length The metric path length is not important metric is as not per important our objectives as per dueour to objectives low success due rates. to Thelow results success are shown rates. The results in Figure are shown 3. in Figure 3. Figure Figure 3. Results 3. Results Alpha-1.5 Alpha-1.5 Problem Problem Success Success Rate; Rate; Solution Solution Length. Length. The or set of experiments were run to obtain optimal set of parameters The or set of experiments were run to obtain optimal set of parameters deterministically deterministically varying varying hybrid hybrid sampler sampler and and adaptive adaptive hybrid hybrid sampler. sampler. Parameter Parameter setting setting allows allows us to set best parameter values and thus achieve best permance. The parameters of deterministically varying sampler are probabilities of different samplers. Each setting is denoted as a tuple (P O, P G, P M ) denoting initial selection probabilities of obstacle-based sampler, Gaussian-based sampler and maximum clearance-based Sampler, respectively. The initial probability of unim-based sampler is taken as P U = 1.0 (P O + P G + P M ). The third configuration in Figure 4a is when probabilities remain constant. A heuristic used in paper was that difficult regions should be sampled first, and focus should n be on easier parts of configuration space, which was intuitively inspired. The reverse of same proposal is tried in fourth configuration of Figure 4a. It can be clearly seen that two techniques are inferior to proposed technique parameter setting. For adaptive sampler, in order to keep testing simple, α O and α G are kept same, while α M is kept such that selection probability of maximum clearance sampler is 0.1. These constants, when multiplied by density, give initial selection probabilities. The results are given in Figure 4. The permance can be furr improved by better parameter settings.

11 configuration fourth configuration space, which of Figure was 4a. intuitively It can be inspired. clearly seen The that reverse two of techniques same proposal are inferior is tried to in proposed fourth configuration technique of parameter Figure 4a. setting. It can For be clearly adaptive seen that sampler, two in techniques order to keep are testing inferior simple, to proposed αo and αg technique are kept parameter same, while setting. αm is For kept such adaptive that sampler, selection in order probability to keep of testing maximum simple, αo clearance and αg sampler are kept is 0.1. same, These while constants, αm is kept when such multiplied that selection by density, probability give of initial maximum selection clearance Electronics probabilities. 2016, sampler 5, The 16 results is 0.1. are These given constants, in Figure when 4. The multiplied permance by density, can be furr give improved initial selection by 11better of 13 probabilities. parameter settings. The results are given in Figure 4. The permance can be furr improved by better parameter settings. Figure Figure Results Results Alpha-1.5 Alpha-1.5 Problem Problem using using deterministically deterministically varying varying sampler sampler ( ( x-axis x-axis Figure value is 4. tuple Results (PO, PG, PM) Alpha-1.5 denoting Problem parameters. using deterministically PU 1.0 (PO PG varying PM). sampler Results ( x-axis value is tuple (P O, P G, P M ) denoting parameters. P U 1.0 (P O + P G + P M ); Results value Alpha-1.5 is a tuple Problem (PO, using PG, PM) denoting adaptive sampler parameters. ( x-axis PU = indicates 1.0 (PO + parameter PG + PM). value). Results Alpha-1.5 Problem using adaptive sampler ( x-axis indicates parameter value). Alpha-1.5 Problem using adaptive sampler ( x-axis indicates parameter value). Furr, permance is also studied different timeout values. Here, timeout is strictly taken Furr, as execution permance time. The is is also experiments studied are different permed timeout values. increasing Here, is execution timeout times. is strictly The taken time. The experiments are permed increasing execution times. The results results as are shown execution in Figure time. 5. The At experiments small timeout are values, permed all of increasing algorithms execution barely produce times. The any are shown in Figure 5. At small timeout values, all of algorithms barely produce any results, while results, are while shown at higher in Figure values 5. y At small nearly timeout always values, produce all of results. algorithms The proposed barely produce approach any is results, at higher values y all nearly always produce results. The proposed approach is clearly clearly while best at higher all values timeout y values, all nearly baring always a single produce case in results. which The differences proposed approach are due is to clearly best all timeout values, baring a single case in which differences are due to invalidity of invalidity of best parameters all timeout set values, particular baring a case. single case in which differences are due to invalidity parameters of set parameters particular set case. particular case. Electronics 2016, 5, of 12 (c) (d) Figure Figure Results Results Alpha-1.5 Alpha-1.5 Problem Problem different different execution execution times times s; s; s; s; (c) (c) s; s; (d) (d) s. s. 6. Conclusions With increasing autonomy in robots, role of motion planning is to quickly compute a path to any goal configuration, while robot may be operating in a variety of complex to simple scenarios. We have presented a hybrid sampling strategy using a number of samplers. First, selection probabilities of samplers were deterministically varied using a pre-computed rule. Then,

12 Electronics 2016, 5, of Conclusions With increasing autonomy in robots, role of motion planning is to quickly compute a path to any goal configuration, while robot may be operating in a variety of complex to simple scenarios. We have presented a hybrid sampling strategy using a number of samplers. First, selection probabilities of samplers were deterministically varied using a pre-computed rule. Then, strategy was made adaptive, which enables selection probabilities to be determined dynamically based on environment. Experiments were done and comparisons with standard sampling techniques were made. Experiments revealed that adaptive technique achieved best permance over metric of percentage of times algorithm could find a solution. The adaptive technique was followed by deterministically varying technique, while basic PRM achieved worst permance. The experiments show a certain advantage of using adaptive hybrid sampling. In future, better ways of making samplers adaptive depending upon temporal permance will be tried. New samplers will be designed to better suit specific requirements. The experiments need to be permed on more benchmarks and better metrics of permance. The experiments need to be permed on real robots. Acknowledgments: The authors express ir sincere thanks to Shiv Dhingra from Indian Institute of Inmation Technology, Allahabad his constant support in development of work. Author Contributions: Prashant Gupta and Rishabh Tiwari conceived and designed algorithms; Ashwin Kannan developed algorithms; Apurv Khatri and Shubham Prasad analyzed algorithms; Prashant Gupta, Rishabh Tiwari, Ashwin Kannan and Rahul Kala wrote paper. Conflicts of Interest: The authors declare no conflict of interest. References 1. Choset, H.; Lynch, K.M.; Hutchinson, S.; Kantor, G.A.; Burgard, W.; Kavraki, L.E.; Thrun, S. Principles of Robot Motion: Theory, Algorithms, and Implementations; MIT Press: Cambridge, MA, USA, Tiwari, R.; Shukla, A.; Kala, R. Intelligent Planning Mobile Robotics: Algorithmic Approaches; IGI Global Publishers: Hershey, PA, USA, De Berg, M.; Cheong, O.; van Kreveld, M.; Overmars, M. Computational Geometry: Algorithms and Applications, 3rd ed.; Springer Verlag: Berlin, Germany, Russell, S.; Norvig, P. Artificial Intelligence: A Modern Approach, 3rd ed.; Pearson: Harlow, UK, Kavraki, L.E.; Svestka, P.; Latombe, J.-C.; Overmars, M.H. Probabilistic roadmaps path planning in high-dimensional configuration spaces. IEEE Trans. Rob. Automat. 1996, 12, [CrossRef] 6. Hsu, D.; Sanchez-Ante, G.; Sun, Z. Hybrid PRM Sampling with a Cost-Sensitive Adaptive Strategy. In Proceedings of 2005 IEEE International Conference on Robotics and Automation, Barcelona, Spain, April 2005; pp Rodriguez, S.; Thomas, S.; Pearce, R.; Amato, N.M. RESAMPL: A Region-Sensitive Adaptive Motion Planner. In Algorithmic Foundation of Robotics VII, Springer Tracts in Advanced Robotics; Springer-Verlag: Berlin/Heidelberg, Germany, 2008; Volume 47, pp Kurniawati, H.; Hsu, D. Workspace-Based Connectivity Oracle: An Adaptive Sampling Strategy PRM Planning. In Algorithmic Foundation of Robotics VII, Springer Tracts in Advanced Robotics; Springer-Verlag: Berlin/Heidelberg, Germany, 2008; Volume 47, pp Morales, M.; Tapia, L.; Pearce, R.; Rodriguez, S.; Amat, N.M. A Machine Learning Approach Feature-Sensitive Motion Planning. In Algorithmic Foundations of Robotics VI, Springer Tracts in Advanced Robotics; Springer-Verlag: Berlin/Heidelberg, Germany, 2005; Volume 17, pp LaValle, S.M.; Kuffner, J.J. Randomized kinodynamic planning. Int. J. Robot. Res. 2001, 20, [CrossRef] 11. Karaman, S.; Frazzoli, E. Sampling-based algorithms optimal motion planning. Int. J. Rob. Res. 2011, 30, [CrossRef]

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