Path Planning And Navigation

Path Planning Techniques Sasi Bhushan Beera

Shreeganesh Sudhindra

Path Planning

• Compute motion strategies, e.g.,– Geometric paths – Time-parameterized trajectories

• Achieve high-level goals, e.g.,– To build a collision free path from start point

to the desired destination– Assemble/disassemble the engine– Map the environment

Objective• To Compute a collision-free path for a mobile robot among static

obstacles.• Inputs required

– Geometry of the robot and of obstacles – Kinematics of the robot (d.o.f)– Initial and goal robot configurations (positions &

orientations)

• Expected Result Continuous sequence of collision-free robot configurations

connecting the initial and goal configurations

Some of the existing Methods• Visibility Graphs

• Roadmap

• Cell Decomposition

• Potential Field

Visibility Graph Method

• If there is a collision-free path between two points, then there is a polygonal path that bends only at the obstacles vertices.

• A polygonal path is a piecewise linear curve.

Visibility Graph

• A visibility graph is a graph such that– Nodes: qinit, qgoal, or an obstacle vertex.– Edges: An edge exists between

nodes u and v if the line segment between u and v is an obstacle edge or it does not intersect the obstacles.

Slide 10

Breadth-First Search

Slide 11


Slide 12


Slide 13


Slide 14


Slide 15


Slide 16


Slide 17


Slide 18


A Simple Algorithm for Building Visibility Graphs

Road mapping Technique

• Visibility graph

• Voronoi Diagram Introduced by computational

geometry researchers.

Generate paths that maximizes clearance

Applicable mostly to 2-D configuration spaces

Slide 21

Cell-decomposition Methods

• Exact cell decomposition• Approximate cell decomposition

– F is represented by a collection of non-overlapping cells whose union is contained in F.

– Cells usually have simple, regular shapes, e.g., rectangles, squares.

– Facilitate hierarchical space decomposition

Slide 22

Quadtree Decomposition

Slide 23

Octree Decomposition

Slide 24

Algorithm Outline

Slide 25

Potential Fields

• Initially proposed for real-time collision avoidance [Khatib 1986].

• A potential field is a scalar function over the free space.

• To navigate, the robot applies a force proportional to the negated gradient of the potential field.

• A navigation function is an ideal potential field that– has global minimum at the goal– has no local minima– grows to infinity near obstacles– is smooth

Slide 26

Attractive & Repulsive Fields

Slide 27

How Does It Work?

Slide 28

Algorithm Outline

• Place a regular grid G over the configuration space• Compute the potential field over G• Search G using a best-first algorithm with potential

field as the heuristic function Note: A heuristic function or simply a heuristic is

a function that ranks alternatives in various search algorithms at each branching step basing on an available information in order to make a decision which branch is to be followed during a search.

Slide 29

Use the Local Minima Information

– Identify the local minima – Build an ideal potential field – navigation

function – that does not have local minima– Using the calculations done so far, we can create

a PATH PLANNER which would give us the optimum path for a set of inputs.

Active Sensing

The main question to answer:”Where to move next?”

Given a current knowledge about the robot state and the environment, how to select the next sensing action or sequence of actions. A vehicle is moving autonomously through an environment gathering information from sensors. The sensor data are used. to generate the robot actions

Beginning from a starting configuration (xs,ys,s) to a goal configuration (xg,yg,g) in the presence of a reference trajectory and without it; With and without obstacles; Taking into account the constraints on the velocity, steering angle, the obstacles, and other constraints …

Active Sensing of a WMRActive Sensing of a WMR

k

ky

kx

kk

k

kkkk

kkkk

k

k

k

LTv

TvyTvx

yx

,

,

,

1

1

1

sin

)sin()cos(

Robot modelyy

x

k

kky

kx

L

Beacon

Bx

By

Trajectory optimizationTrajectory optimization • Between two points there are an infinite number of possible Between two points there are an infinite number of possible

trajectories. But not each trajectory from the configuration space trajectories. But not each trajectory from the configuration space represents a feasible trajectory for the robot. represents a feasible trajectory for the robot.

• How to move in the best way according to a criterion from the How to move in the best way according to a criterion from the starting to a goal configuration?starting to a goal configuration?

• The key idea is to use some parameterized family of possible The key idea is to use some parameterized family of possible trajectories and thus to reduce the infinite-dimensional problem to a trajectories and thus to reduce the infinite-dimensional problem to a finitely parametrized optimization problem. To characterize the robot finitely parametrized optimization problem. To characterize the robot motion and to process the sensor information in efficient way, an motion and to process the sensor information in efficient way, an appropriate criterion is need. So, active sensing is a decision appropriate criterion is need. So, active sensing is a decision making, global optimization problem subject to constraintsmaking, global optimization problem subject to constraints..

Trajectory OptimizationTrajectory Optimization• Let Let Q is a class of smooth functions. The problem of determining is a class of smooth functions. The problem of determining

the ‘best’ trajectory the ‘best’ trajectory qq with respect to a criterion with respect to a criterion J J can be then can be then formulated as formulated as

q = argmin(J)q = argmin(J)where the owhere the optimization criterionptimization criterion is chosen of the form is chosen of the form

information part losses (time, traveled distance)information part losses (time, traveled distance) subject to subject to constraintsconstraints ll: lateral deviation, v: WMR velocity; : lateral deviation, v: WMR velocity;

: steering angle; d : distance : steering angle; d : distance to obstacleto obstacle

CccJkiA

21,

min I

,,,, min,,maxmaxmax,, okokkyky ddvvll

Trajectory OptimizationTrajectory OptimizationThe class The class Q of harmonic functions is chosen,

Q = Q(p),

pp: : vector of parameters obeying to preset constraints;Given N number of harmonic functions, the new modified robot

trajectory is generated on the basis of the reference trajectory by a lateral deviation as a linear superposition

Why harmonic functions?• They are smooth periodic functions;• Gives the possibility to move easily the robot to the desired final

point;• Easy to implement;• Multisinusoidal signals are reach excitation signals and often used

in the experimental identification. They have proved advantages for control generation of nonholonomic WMR (assure smooth stabilization). For canonical chained systems Brockett (1981) showed that optimal inputs are sinusoids at integrally related frequencies, namely 2, 2. 2, …, m/2. 2.

Optimality CriterionOptimality Criterion

I = trace(WP), I = trace(WP),

I is computed at the goal configuration or on the the whole I is computed at the goal configuration or on the the whole trajectory (part of it, e.g. in an interval ) trajectory (part of it, e.g. in an interval )

where W = MN;where W = MN;M: scaling matrix; N: normalizing matrixM: scaling matrix; N: normalizing matrixP: estimation error covariance matrix (information matrix P: estimation error covariance matrix (information matrix

or entropy) from a filter (EKF); or entropy) from a filter (EKF);

],[ ba kk

CccJkiA

21,

min I

Lyudmila Mihaylova, Katholieke Universiteit Leuven, Division PMA

Trajectory (N=2 sinusoids)

Beacon

N=0

N=1

0 2 4 6 8 10 1214

15

16

17

18

19y

posit

ion,

[m]

x position, [m]

Beacon

N=0

N=2

Point-to-point optimization

0 2 4 6 8 10 12

12

13

14

15

16

17

18

19

20

21y

posit

ion,

[m]

x position, [m]

BeaconStart

Goal

Obstacle

N=3

Lyudmila Mihaylova, Katholieke Universiteit Leuven, Division PMA

Implementation

• Using Optimization Toolbox of MATLAB, • fmincon finds the constrained minimum of a function of

several variables• With small number of sinusoids (N<5) the

computational complexity is such that it is easily implemented on-line. With more sinusoidal terms (N>10), the complexity (time, number of computations) is growing up and a powerful computer is required or off-line computation. All the performed experiments prove that the trajectories generated even with N=3 sinusoidal terms respond to the imposed requirements.

Conclusions

An effective approach for trajectories optimization has been considered :

• Appropriate optimality criteria are defined. The influence of the different factors is decoupled;

• The approach is applicable in the presence of and without obstacles.

Few Topics with related videos of work done by people around the world in the field Robot navigation

• 3-D path planning and target trajectory prediction : http://www.youtube.com/watch?v=pr1Y21mexzs&feature=related

• Path Planning : http://www.youtube.com/watch?v=d0PluQz5IuQ

• Potential Function Method – By Leng Feng Lee : http://www.youtube.com/watch?v=Lf7_ve83UhE

• Real-Time Scalable Motion Planning for Crowds -http://www.youtube.com/watch?v=ifimWFs5-hc&NR=1

• Robot Potential with local minimum avoidance -http://www.youtube.com/watch?v=Cr7PSr6SHTI&feature=related

References• Tracking, Motion Generation and Active Sensing of

Nonholonomic Wheeled Mobile Robots -Lyudmila Mihaylova & Katholieke Universiteit Leuven

• Robot Path Planning - By William RegliDepartment of Computer Science(and Departments of ECE and MEM)Drexel University

• Part II-Motion Planning- by Steven M. LaValle (University of Illinois)

• Wikipedia• www.mathworks.com

Technology

Path Planning And Navigation