plannerRRTStar

Create an optimal RRT path planner (RRT*)

Description

The plannerRRTStar object creates an asymptotically-optimal RRT planner, RRT*. The RRT* algorithm converges to an optimal solution in terms of the state space distance. Also, its runtime is a constant factor of the runtime of the RRT algorithm. RRT* is used to solve geometric planning problems. A geometric planning problem requires that any two random states drawn from the state space can be connected.

Creation

Syntax

planner = plannerRRTStar(stateSpace,stateVal)

planner = plannerRRTStar(___,Name=Value)

Description

planner = plannerRRTStar(stateSpace,stateVal) creates an RRT* planner from a state space object, stateSpace, and a state validator object, stateVal. The state space of stateVal must be the same as stateSpace. stateSpace and stateVal also sets the StateSpace and StateValidator properties of the planner object.

planner = plannerRRTStar(___,Name=Value) sets properties using one or more name-value arguments in addition to the input arguments in the previous syntax. You can specify the StateSampler, BallRadiusConstant, ContinueAfterGoalReached, MaxNumTreeNodes, MaxIterations, MaxConnectionDistance, GoalReachedFcn, and GoalBias properties as name-value arguments.

Properties

StateSpace — State space for planner

state space object

State space for the planner, specified as a state space object. You can use state space objects such as stateSpaceSE2, stateSpaceDubins, stateSpaceReedsShepp, and stateSpaceSE3. You can also customize a state space object using the nav.StateSpace object.

StateValidator — State validator for planner

state validator object

State validator for the planner, specified as a state validator object. You can use state validator objects such as validatorOccupancyMap, validatorVehicleCostmap, and validatorOccupancyMap3D.

StateSampler — State space sampler for sampling input space

stateSamplerUniform object (default) | stateSamplerGaussian object | stateSamplerMPNET object | nav.StateSampler object

Since R2023b

State space sampler used for finding state samples in the input space, specified as a stateSamplerUniform object, stateSamplerGaussian object, stateSamplerMPNET object, or nav.StateSampler object. By default, the plannerRRTStar uses uniform state sampling.

BallRadiusConstant — Constant used to estimate the near neighbors search radius

100 (default) | positive scalar

Constant used to estimate the near neighbors search radius, specified as a positive scalar. The radius is estimated as following:

$$r=\min \left(\gamma {\left(\frac{\log (n)}{n}\right)}^{\frac{1}{d}},\eta \right)$$

where:

• γ — The value of the BallRadiusConstant property

• n — Current number of nodes in the tree

• d — Dimension of the state space

• η — The value of the MaxConnectionDistance property

γ is defined as following:

$$\gamma ={{\displaystyle 2}}^d\left(1+\frac{1}{d}\right)\left(\frac{{{\displaystyle V}}_{Free}}{{{\displaystyle V}}_{Ball}}\right)$$

where:

• V_Free — Approximate free volume in search-space

• V_Ball — Volume of unit ball in d dimensions

The formulae above define a BallRadiusConstant of "appropriate" size for a given space, meaning that as the number of nodes filling the space grows and the radius shrinks, the expected number of neighbors grows logarithmically. Higher values will result in a higher average number of neighbors within the d-ball per iteration, leading to more rewire candidates. However, values below this suggested minimum could lead to a single nearby neighbor, which fails to produce asymptotically optimal results.

Example: BallRadiusConstant=80

Data Types: single | double

ContinueAfterGoalReached — Continue to optimize after goal is reached

false (default) | true

Decide if the planner continues to optimize after the goal is reached, specified as false or true. The planner also terminates regardless of the value of this property if the maximum number of iterations or maximum number of tree nodes is reached.

Example: ContinueAfterGoalReached=true

Data Types: logical

MaxNumTreeNodes — Maximum number of nodes in the search tree

1e4 (default) | positive integer

Maximum number of nodes in the search tree (excluding the root node), specified as a positive integer.

Example: MaxNumTreeNodes=2500

Data Types: single | double

MaxIterations — Maximum number of iterations

1e4 (default) | positive integer

Maximum number of iterations, specified as a positive integer.

Example: MaxIterations=2500

Data Types: single | double

MaxConnectionDistance — Maximum length of motion

0.1 (default) | positive scalar

Maximum length of a motion allowed in the tree, specified as a scalar.

Example: MaxConnectionDistance=0.3

Data Types: single | double

GoalReachedFcn — Callback function to determine whether goal is reached

@nav.algs.checkIfGoalIsReached | function handle

Callback function to determine whether the goal is reached, specified as a function handle. You can create your own goal reached function. The function must follow this syntax:

 function isReached = myGoalReachedFcn(planner,currentState,goalState)

where:

• planner — The created planner object, specified as plannerRRTStar object.

• currentState — The current state, specified as a three element real vector.

• goalState — The goal state, specified as a three element real vector.

• isReached — A boolean variable to indicate whether the current state has reached the goal state, returned as true or false.

To use custom GoalReachedFcn in code generation workflow, this property must be set to a custom function handle before calling the plan function and it cannot be changed after initialization.

Data Types: function handle

GoalBias — Probability of choosing goal state during state sampling

0.05 (default) | real scalar in range [0,1]

Probability of choosing the goal state during state sampling, specified as a real scalar in range [0,1]. The property defines the probability of choosing the actual goal state during the process of randomly selecting states from the state space. You can start by setting the probability to a small value such as 0.05.

Example: GoalBias=0.1

Data Types: single | double

Object Functions

plan	Plan path between two states
copy	Create copy of planner object

Examples

Plan Optimal Path Between Two States

Create a state space.

ss = stateSpaceSE2;

Create an occupancyMap-based state validator using the created state space.

sv = validatorOccupancyMap(ss);

Create an occupancy map from an example map and set map resolution as 10 cells/meter.

load exampleMaps.mat
map = occupancyMap(simpleMap,10);
sv.Map = map;

Set validation distance for the validator.

sv.ValidationDistance = 0.01;

Update state space bounds to be the same as map limits.

ss.StateBounds = [map.XWorldLimits; map.YWorldLimits; [-pi pi]];

Create RRT* path planner and allow further optimization after goal is reached. Reduce the maximum iterations and increase the maximum connection distance.

planner = plannerRRTStar(ss,sv, ...
          ContinueAfterGoalReached=true, ...
          MaxIterations=2500, ...
          MaxConnectionDistance=0.3);

Set the start and goal states.

start = [0.5 0.5 0];
goal = [2.5 0.2 0];

Plan a path with default settings.

rng(100,'twister') % repeatable result
[pthObj,solnInfo] = plan(planner,start,goal);

Visualize the results.

map.show
hold on
% Tree expansion
plot(solnInfo.TreeData(:,1),solnInfo.TreeData(:,2),'.-')
% Draw path
plot(pthObj.States(:,1),pthObj.States(:,2),'r-','LineWidth',2)

Plan Path Through 3-D Occupancy Map Using RRT Star Planner

Load a 3-D occupancy map of a city block into the workspace. Specify the threshold to consider cells as obstacle-free.

mapData = load("dMapCityBlock.mat");
omap = mapData.omap;
omap.FreeThreshold = 0.5;

Inflate the occupancy map to add a buffer zone for safe operation around the obstacles.

inflate(omap,1)

Create an SE(3) state space object with bounds for state variables.

ss = stateSpaceSE3([0 220;0 220;0 100;inf inf;inf inf;inf inf;inf inf]);

Create a 3-D occupancy map state validator using the created state space. Assign the occupancy map to the state validator object. Specify the sampling distance interval.

sv = validatorOccupancyMap3D(ss, ...
     Map = omap, ...
     ValidationDistance = 0.1);

Create a RRT star path planner with increased maximum connection distance and reduced maximum number of iterations. Specify a custom goal function that determines that a path reaches the goal if the Euclidean distance to the target is below a threshold of 1 meter.

planner = plannerRRTStar(ss,sv, ...
          MaxConnectionDistance = 50, ...
          MaxIterations = 1000, ...
          GoalReachedFcn = @(~,s,g)(norm(s(1:3)-g(1:3))<1), ...
          GoalBias = 0.1);

Specify start and goal poses.

start = [40 180 25 0.7 0.2 0 0.1];
goal = [150 33 35 0.3 0 0.1 0.6];

Configure the random number generator for repeatable result.

rng(1,"twister");

Plan the path.

[pthObj,solnInfo] = plan(planner,start,goal);

Visualize the planned path.

show(omap)
axis equal
view([-10 55])
hold on
% Start state
scatter3(start(1,1),start(1,2),start(1,3),"g","filled")
% Goal state
scatter3(goal(1,1),goal(1,2),goal(1,3),"r","filled")
% Path
plot3(pthObj.States(:,1),pthObj.States(:,2),pthObj.States(:,3), ...
      "r-",LineWidth=2)

More About

Ball Radius Constant

The major difference between RRT and RRT* is the rewiring behavior, which guarantees asymptotic optimality for the RRT* algorithm. When RRT-based planners generate a new node, the planner finds the nearest node in the tree. If the path between nodes is collision free and otherwise valid, the RRT algorithm connects the nodes, but the RRT* algorithm performs additional steps to optimize the tree after connecting the nodes. First, RRT* finds all nodes in the tree within some distance to the new node, then RRT* finds the node that provides the new node with the shortest valid path back to the starting node, and adds an edge between the node and the new node. Lastly, the planner performs a rewire operation, which checks whether the new node can provide each of the nearest nodes with a shorter route back to the starting node. In the case that there is a shorter path, the node is disconnected from the current parent and reparented to the closer node.

The radius at which the rewiring occurs, is called the ball radius constant. Selecting an appropriate ball radius constant is important as the goal of RRT* is to guarantee asymptotic optimality while limiting any additional overhead computation. If the ball radius constant is too large, the runtime of RRT* increases. If the ball radius constant is too small, then the algorithm may fail to converge on an optimal result.

plannerRRTStar uses this distance formula, adapted from [1], to find the nearest neighbors:

$$r=\min \left(\gamma {\left(\frac{\log (n)}{n}\right)}^{\frac{1}{d}},\eta \right)$$

where:

• n — Number of nodes in tree

• d — Number of dimensions of the state space

• η — Maximum connection distance

• γ — Ball radius constant, defined as:

  $$\gamma ={{\displaystyle 2}}^d\left(1+\frac{1}{d}\right)\left(\frac{{{\displaystyle V}}_{Free}}{{{\displaystyle V}}_{Ball}}\right)$$

  where:

  • V_Free — Lebesgue Measure, the approximate free volume in search space

  • V_Ball — Volume of unit ball in d dimensions

Meaning and Intuition of Ball Radius Constant

The formulae for r and γ define a radius of appropriate size for a given space and sampling density. And as the number of nodes filling the space grows linearly, the radius must shrink and the number of neighbors inside the shrinking ball grows logarithmically.

This intuition is from the expectation that all the newly sampled points in the tree have been uniformly and independently sampled from a free portion of the configuration space. By sampling points in this way, you can say they were generated using a homogeneous Poisson point process. This means that in each iteration of RRT*, n points have been uniformly sampled in the free space, so there should be an average density of points per unit volume, λ. For spaces of arbitrary dimensions, there is an intensity of points per unit measure.

$$\lambda =\frac{N}{V_{Free}}$$

Therefore, the number of points you can expect to see in any portion of the planning space is the volume of that portion, multiplied by the density. For RRT*, the focus is in the number of points within a ball of d dimensions of radius r:

$$n_{1,d}=V_{Ball}·\lambda =\frac{V_{Ball}}{V_{Free}}·n$$

$${\text{n}}_{1,d}={\text{n}}_{1,d}·r^d$$

where,

• n_1,d — Expected number of points inside unit ball of d dimensions

• n_r,d — Expected number of points inside ball of d dimensions with a radius r

And recalling that the goal for the number of neighbors is to grow logarithmically as n approaches infinity, you can set n_r,d=log(n) and solve for r:

$$\text{r=}{\left(\frac{V_{Free}}{V_{Ball}}·\frac{\log (n)}{n}\right)}^{\frac{1}{d}}$$

The remaining coefficients from formula 2 are derived from the convergence proof in [1]. However with n removed you can see that the ball radius constant is a ratio of the free space in the sample region vs the measure of the unit ball multiplied by a dimension-specific constant.

References

[1] Karaman, S. and E. Frazzoli. "Sampling-Based Algorithms for Optimal Motion Planning." The International Journal of Robotics Research. Vol. 30, Number 7, 2011, pp 846 – 894.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• To use custom GoalReachedFcn in code generation workflow, this property must be set to a custom function handle before calling the plan function and it cannot be changed after initialization.

Version History

Introduced in R2019b

R2023b: Specify sampling approach for path planning

You can now specify uniform sampling, Gaussian sampling, MPNet sampling, or a custom sampling approach to generate samples for path planning. Use the name, value argument StateSampler to specify the sampling approach.