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
Description
creates an RRT* planner from a state space object, planner
= plannerRRTStar(stateSpace
,stateVal
)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.
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.planner
= plannerRRTStar(___,Name=Value
)
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:
where:
γ — The value of the
BallRadiusConstant
propertyn — 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:
where:
VFree — Approximate free volume in search-space
VBall — 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 asplannerRRTStar
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 astrue
orfalse
.
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
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:
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:
where:
VFree — Lebesgue Measure, the approximate free volume in search space
VBall — Volume of unit ball in d dimensions
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.
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:
where,
n1,d — Expected number of points inside unit ball of d dimensions
nr,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 nr,d=log(n) and solve for r:
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 R2019bR2023b: 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.
See Also
Objects
plannerRRT
|plannerBiRRT
|stateSpaceReedsShepp
|stateSpaceDubins
|stateSpaceSE2
|stateSpaceSE3
|validatorOccupancyMap
|validatorVehicleCostmap
|validatorOccupancyMap3D
|stateSamplerUniform
Functions
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