Dynamic Path Planning
FOLLOW THE GAP METHOD [FGM] FOR MOBILE ROBOTS
Presented by: Vikrant Kumar M. Tech. MED CIM 133569
Robotics – Control & Intelligence – Path
Planning – Dynamic Path Planning
Controller Design Sensors for Robot
Mobile Robot Navigation
• Global Navigation – from knowledge of goal point
• Local Navigation – from knowledge of near by objects
• Personal Navigation – continuous updating of current
Robot’s ability to safely move towards the Goal using its knowledge and
sensorial information of the surrounding environment.
Three terms important in navigation are:
Static Path Planning
• Probabilistic Roadmap (PRM) - Two phase navigation:
• Learning phase
• Query phase
• Visibility Graph – navigating at the boundary of obstacles,
turning at corners only, finding shortest straight line path.
Based on a map and goal location, finding a geometric path.
Dynamic Path Planning
• Bug Algorithms
• Artificial Potential Field (APF) Algorithm
• Harmonic Potential Field (HPF) Algorithm
• Virtual Force Field (VFF) method
• Virtual Field Histogram (VFH) method
• Follow the Gap Method (FGM)
Aim is of avoiding unexpected obstacles along the robot’s trajectory to reach the goal.
Some terms of concern
• Point Robot Approach
• Field of view of Robot
• Non-holonomic constraints
Point Robot Approach
• Robot and Obstacles are assumed circular.
• Radius of robot is added to radius of obstacles
• The Robot is reduced to a point, while Obstacles are equally enlarged.
Field of view
• The sector region within the range of robot’s sensors to get
information of environment.
• Two quantitative measures of field of view:
• End angles of the sector on right and left sides.
• Radius of the sector.
• If the vector space of the possible motion directions of a mechanical
system is restricted
• And the restriction can not be converted into an algebraic relation
between configuration variables.
• Can be visualized as, inability of a car like vehicle to move sideways, it
is bound to follow an arc to reach a lateral co-ordinate.
Nonholonomic Constraints and Field of View
Field of View
• Common sense approach of moving directly to goal.
• Contour the obstacle when found, until moving straight to goal is
• Path chosen – often too long
• Robot prone to move close to obstacles
• Main drawback –
• Robot gets trapped in local minima.
• The Method Ignores nonholonomic constraints
Harmonic Potential Field (HPF)
• An HPF is generated using a Laplace boundary value problem (BVP).
• HPF approach may be configured to operate in a model-based and/or
• It can also be made to accommodate a variety of constraints.
• the robot must know the map of the whole environment .
• contradicts reactiveness and local planning properties of obstacle
Virtual Force Field method (VFF)
• 2D Cartesian histogram grid for obstacle representation.
• Each cell has certainty value of confidence, that an obstacle is present there.
• Then APF is applied.
• Problems of APF method still exist in VFF
Virtual Field Histogram (VFH)
• Uses a 2D Cartesian histogram grid like in VFF.
• Reduces it to a one dimensional polar histogram around the robot's
• Selects lowest polar obstacle density sector
• steers the robot in that direction
• very much goal oriented since it always selects the sector which is in the
same direction as the goal.
• selected sector can be the wrong one in some cases.
• does not consider nonholonomic constraints of robots
Follow the Gap Method (FGM)
• Point Robot Approach
• Obstacle representation
• Construction a gap array among obstacles.
• Determination of maximum gap, considering the Goal point location.
• Calculation of angle to Center of Maximum gap
• Robot proceeds to center of maximum gap.
• The Algorithm
• Should find a purely reactive heading to achieve goal co-ordinates
• Should avoiding obstacles with as large distance as possible
• Should consider measurement and nonholonomic constraints
• for obstacle avoidance must collaborate with global planner
• Goal point – obtained from the global planner
• Obstacle co-ordinates - change with time
Point Robot Approach
Xrob = Abscissa of robot point
Yrob = Ordinate of robot point
Rrob = Robot circle’s radius
Xobsn = Abscissa of nth obstacle
Yobsn = Ordinate of nth obstacle
Robsn = nth obstacle’s circle’s radius
Distance to Obstacle
Distance of nth obstacle from robot
d = ((Xobsn – Xrob)2 + (Yobsn – Yrob)2)1/2
Using Pythogoras theorem
dn2 + (Robsn + Rrob)2 = d2
Or, dn = ((Xobsn – Xrob)2 + (Yobsn – Yrob)2 – (Robsn + Rrob)2)1/2
• Two parameter representation
• Φ obs_l_1 – Border left angle of obstacle 1
• Φ obs_r_1 -- Border right angle of obstacle 1
• Φ obs_l_1 – Border left angle of obstacle 2
• Φ obs_l_1 – Border right angle of obstacle 2
Gap Border Evaluation
If, 𝑑𝑛ℎ𝑜𝑙 < 𝑑𝑓𝑜𝑣 => 𝛷𝑙𝑖𝑚 = 𝛷𝑛ℎ𝑜𝑙
Else if, 𝑑𝑛ℎ𝑜𝑙 ≥ 𝑑𝑓𝑜𝑣 => 𝛷𝑙𝑖𝑚 = 𝛷𝑓𝑜𝑣
In order to understand which boundary is active for a
boundary obstacle, decision rule are illustrated as
Gap boarder parameters
• 1. Φlim: Gap border angle
• 2. Φnhol: Border angle coming from nonholonomic constraint
• 3. Φfov: Border angle coming from field of view
• 4. dnhol: Nearest distance between nonholonomic constraint arc and
• 5. dfov: Nearest distance between field of view line and obstacle border
Gap center angle
• The gap center angle (φgap_c ) is found in terms of the measurable d1,
d2, φ1, φ2 parameters
Calculation of final heading angle
• Final angle is Combination of angle of center of maximum gap and
Goal point angle.
• Determined by fusing weighted average function of gap center angle
and goal angle.
• α is the weight to obstacle gap.
• α acts as tuning parameter for FGM.
• ß weight to goal point (assumed 1 for simplicity)
• dmin is minimum distance to the approaching obstacle.
Role of α value
• Weightage to gap angle is α/dmin
• α makes the path goal oriented or gap oriented.
• For α= 0, φfinal is equal to φgoal
• Increasing values of alpha brings φfinal closer to φgap_c and vice
FGM and APF on local minima
• FGM the robot can reach goal point while avoiding obstacles
• In APF method, robot gets stuck because of the local minimum where
all vectors from the obstacles and goal point zero each other
• FGM selects the first calculated gap value if there are equal maximum
• This provides FGM to move if at least one gap exists.
Comparison of Safety and Travel length
• From table below, FGM is 23% safer than the FGM-basic and 40%
safer than the APF in terms of the norm of the defined metric while
the total distance traveled values are almost the same
Dead end Scenario
• A dead-end scenario of U-shaped obstacles is a problem for FGM as it
is for APF as both are more sort of local planners.
• It needs upper level of intelligence.
• Can be solved by approaches like Virtual Obstacle Method, Multiple
Goal Point method etc.
Advantages of FGM
• Single tuning parameter (α) in weightage to gap center angle
• No local minima problem like earlier algorithms
• Considers nonholonomic constraints for the robot.
• Only feasible trajectories are generated, lesser ambiguity to decision,
lesser computation time.
• Field of view of robot is taken into account.
• Robot does not move in unmeasured directions.
• Passage through maximum gap center – Safest path.
Limitation of FGM
• Unable to come out of dead-end-scenario
• Hybridizing FGM with local planner techniques like virtual
obstacles, virtual goal point method etc.
• Dynamic path planning literature and algorithms were explained.
• Follow the Gap Method(FGM) was explained in detail.
• Major Contribution from FGM:
• Single tuning parameter
• No local minima problem
• Consideration to field of view and nonholonomic constraints.
• Consideration to safety in trajectory planning.