Samplin-based Motion Planners

Motion Planning in Robotics

Visual SLAM - An Introduction

Simulataneous Localization and Mapping - An Introduction


Path and trajectory are two very commong terms in robotics, mostly used during motion planning .

Given the complexity of a common robot operational indoor/outdoor scene, the ideal expectation of a motion planning algorithm functional across all possible scenarios is extremely challenging. While there is enough effort put into exploiting the robot's physical model and degrees of freedom during motion planning; there is substantial effort put into modeling the environment and its constraints as well.

For instance, navigation of a mobile robot (assumed to be a point object located at the robot's geometrical center ) in a warehouse involves having a padding (generally equal to the robot footprint) around all the edges of the warehouse and around the obstacles because it is practically impossible for the robot's center to go further out. Similarly, an industrial manipulator arm with fencing all around cannot obtain a pose where, though the end-effector lies within the allowed workspace has an IK configuration with a portion of the robot extruding out of the fencing.

Such intricacies necissate the formulation of different motion planning algorithms with varying assumptions and performance specifications. This article discusses the sampling-based motion planning techniques and its variants, the most used techniques implemented on mobile robots used in the industry and academia alike.

Figure 1 - Mobile Robot Navigation Scene with Padding


MP algorithms are generally designed knowing the limitations and demands of the environment. Also, a lot of motion planning attempts to reduce the environment and obtain a simplified version of the same for computational interpretation. Discrete search techniques are used to derive finite motion waypoints that connect the start and end. A grid-based representation of the environment is one such example, which, although promises optimality and quick solution, it is neither an adequate representation of the environment nor suitable for high dimensional state-space.

Instead of systematic discretization of the C-space and employing search algorithms, sampling-based algorithms randomly extract samples from the C-space and then construct a path out of it. Sampling-based algorithms are more useful in high-dimensional scenarios and find more optimal solutions.

Motion planning eventually is a PSPACE-hard problem where the complexity grows exponentially with C-space dimensions and gets extremely challenging with completeness and optimality requirements.

Figure 2 - Warehouse Map - Grid Representation of the Environment

Discrete Motion Planning

It is important to acknowledge the discrete motion planning pipeline and its nuances. Despite the already mentioned limitations, discrete MP is still employed on several ocassions for ease of use and in limited complexity applications.

Discrete-search creates a discrete, finite, systematic and specific quantizated representation of the environment, obtain action-space and their involved costs and eventually employ the concerned search algorithm to find the path. They generally employ techniques like Breadth-First search, Depth-First search, A* and its variants and Dijkstra algorithms to find paths for the robot.

Figure 3 - Discrete Motion Planning Pipeline

Figure 4 - Depth-First & Breadth-First Search

Owing to the exploding nature of runtime and computational expense of search algorithms for large discrete spaces, dimensionality issues and accrual of potential inaccuracies due to the resolution of the discrete spaces; discrete motion planning becomes a non-ideal, very limited in scope technique.

Sampling-based Motion Planning

Sampling in motion planning uses the complete continuous C-space, draws samples out of it, checks the viability of the sample and eventually tries to use it to create a path towards the goal. Several assumptions and hand-crafted constraints/relaxations on performance and results help in designing very efficient real-time paths for robots.

Sampling is not affected by dimensionality of the C-space and with relaxed completeness (probabilistic completeness, i.e. asymptotic convergence) and sub-optimality conditions, it promises to be the most effective in almost all use-cases.

The Monte-Carlo methods engendered the belief in using a subset instead of all the possibilities in any state-space for search problems. Sampling-based algorithms promise better runtime performance and thus trump other more exhaustive techniques.

Advantages Disadvantages
Probabilistically Complete Unlike to sample narrow passages
Applicable to High Dimensional State Space Sub-optimal solutions
Advantages & Disadvantages of Sampling Based Algorithms

Figure 5 - Sampling-based Motion Planning

The most common sampling-based algorithms discussed here are Probabilistic Roadmaps and Randomly-Exploring Random Trees. There have been several variations proposed and used for these algorithms that have improved performance, completeness, speed and accuracy.


Probabilistic Roadmap planning is a construct and multi-query motion planning technique proposed first in 1996. It has two steps - a learning phase (generally preprocessed ) and a query phase. The learning phase does the bulk work of understanding the workspace upfront before the second query phase which merely searches through the representation derived in the prior phase to provide a final solution.


In the learning phase - several samples are drawn from the workspace and connected to ones nearby, thus creating a roadmap between them all, including the start and desired end point. It lays the foundation for connectivity in the in the Cfree. The learning phase has a construction phase and an expansion phase.

The construction phase creates the roadmap and the expansion phase attempts at filling the gaps in connectivity between sections of the workspace positioned uniquely, involving additional sampling and connections thereafter between the disconnected components.

Learning Phase - Part I - Construction

Figure 6 - Missing Connections during the Construction Phase

Learning Phase - Part II - Expansion

Figure 7 - Enhanced Connectivity in Expansion Phase


The Query Phase is a relatively easier phase with all the bulk computational processing already done. It accepts a start s and a goal g configuration and attempts to find a path between them.

Ideally, a path exists in the roadmap connecting the two and the query returns that path (a collection of all intermediate edges passing through other intermittent nodes that eventually establish connectivity between s and g)

Given that there are several parameters, assumptions and challenges like number of samples, number of retries, sampling techniques, local planners, narrow passages in the map and sampling near obstacles; there are chances of the query failing.

However, limited connectivity in the roadmap and all problems are attempted to be resolved by retrying Learning Phase, exhaustively running Expansion Phase and concurrently operational Learning & Query Phases.

There are several enhanced PRM techniques like Obstacle-Based PRM, Medial-Axis PRM and Simplified PRM among others used to address specific challenges for sampling near obstacles, sampling in narrow passages and sampling problems in general.


Rapidly-Exploring Random Trees (RRT) is the most famous family of sampling-based motion planning algorithms. While PRMs or Potential Field methods are probabilistic in nature and have limitations with substantial effect on planning, RRTs can solve better for lots of constraints.

The algorithm basically starts at some location in the map and starts branching out in random directions, sampling new points at pre-defined distance from the initial location. After an edge is established between the initial point and the new sampled point, the latter becomes the initial location for the next step of branching out.

Using appropriate values for step size, number of sampels to drawn, initial point and other parameters, a densely connected tree-structured map is promised.

The Algorithm

Subtle Features of RRTs

Given the advantages of the basic RRT algorithm, several enhancements like Bidirectional RRT, RRT*, RRT-Connect and RRT*-Smart among others have been used to optimize the solutions and get better performance.

Recently, lots of efforts have been put into using RRT with better hardware (like GPUs), using other search algorithms in conjunction and hand-crafted optimizations for certain operational constraints/desires have fetched roboticists enhanced performance and usage.


Copyright @Akshay Kumar | Last Updated on 05/25/2019