Planar TimeOptimal and LengthOptimal Paths under Acceleration Constraints

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Planar Time-Optimal and Length-Optimal Paths
under Acceleration Constraints

• Aneesh Venkatraman
• Dual Degree Project
• Under the Guidance of
• Prof. Sanjay. P. Bhat
• Department of Aerospace Engineering, IIT Bombay.

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OUTLINE

• Introduction
• The Pontryagin Maximum Principle
• Time-Optimal Paths
• Length-Optimal Paths
• Comparison of Results
• Conclusion

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INTRODUCTION

• Optimal Path Planning in a 2-D plane
• Typical applications
• Optimal horizontal guidance of an aircraft in
  cruise flight.
• Planning the optimal path for a UAV moving in a
  2-D plane in the presence of obstacles.
• Planning the optimal path for an autonomous
  ground vehicle or a robot to reach a target
  point.
• Problem
• To plan the quickest and shortest 2-D path
• from a given initial position and velocity to a
  specified terminal condition
• under a magnitude constraint on the acceleration
  or the turn rate.

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Dubins Problem

• Finding the shortest path for a vehicle moving in
  a 2-D plane from a given initial position and
  heading to a specified terminal position and
  heading such that
• the speed of the vehicle is a constant.
• turn rate is constrained in magnitude.
• The vehicle can move only in the forward
  direction.
• Paths consist of three segments, the first and
  third being circular arcs of minimum turn radius
  and the second one being either a straight line
  or a circular arc of an angle greater than 180
  degrees.

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Dubins Paths

• C refers to circular arc of minimum turn radius
  and S refers to line segment.

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CSC
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Reeds Shepps Problem

• Extension of Dubins work with the vehicle being
  allowed to move both forward and backward.
• Optimal paths are concatenations of straight
  lines and circular arcs of minimum turn radius
  with reversals or equivalently cusps being
  allowed between arcs or line segments unlike
  Dubins case.
• Reeds-Shepps path is shorter than the Dubins
  path.

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Other Problems considered

• D. A. Anisi, J. Hamburg and X. Hu Nearly
  time-optimal but continuous curvature paths for a
  ground vehicle unlike Dubins and Reeds Shepps
  paths by penalizing rapid variation in the
  acceleration.
• H. Erzberger and H. Q. Lee Optimum horizontal
  guidance laws for an aircraft in cruise flight
  moving with a constant velocity.
• G. Yang and V. Kapila Shortest paths for a UAV
  that moves in a 2-D plane with constant velocity
  under a constraint on the minimum turn radius and
  additional constraints arising due to obstacles
  in the path.
• I. Shapira, J. Z. Ben-Asher Optimal path for an
  air vehicle whose velocity magnitude changes
  instantaneously as a linear function of the turn
  rate.

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Previous work

• Constant speed assumption made.
• Time-optimal paths and length-optimal paths are
  the same.
• Paths known to be concatenation of circular arcs
  of minimum turn radius and line segments.
• Present problem
• Velocity changing both in magnitude and
  direction.
• Time-optimal paths and length-optimal paths are
  not the same.
• Solution is not obvious as the vehicle now has
  to choose an optimal combination of accelerating
  forward or decelerating or turning.

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The Pontryagin Maximum Principle

• System
• Terminal Constraints
• Performance Index
• where is to be minimized.
• Hamiltonian

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• The Pontryagin Maximum Principle
• Necessary Conditions for Optimality
• Transversality conditions

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Time-Optimal Paths

• Consider the problem of a vehicle moving in a
  2-D plane from a
• given initial point and velocity to a specified
  terminal condition in
• the minimum possible time under a magnitude
  constraint on
• acceleration.
• Kinematics
• Problem Statement Find the time history for
  such that
• the system is transferred from to
  a specified terminal condition.
• in the minimum possible time .

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Necessary Conditions for Time-Optimality

• Hamiltonian
• Necessary Conditions

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Fixed Terminal Heading, Free Terminal position

• Terminal Condition
• Transversality Conditions
• From the necessary and transversality conditions,
• is a constant for all
• Transversality conditions are satisfied if
• Optimal Trajectory terminates in rest
• Optimal Trajectory does not terminate in rest

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Fixed Terminal Heading, Free Terminal Position

• Optimal Trajectory terminating in rest
• Trajectory ends in rest only if the required
  heading change is obtuse.
• The optimal acceleration vector points opposite
  to the initial velocity vector.
• Optimal paths are straight lines.
• Optimal Trajectory not terminating in rest
• Trajectory does not end in rest only if the
  required heading change is acute.
• The optimal acceleration is the negative of the
  component of the initial velocity vector,
  perpendicular to the final velocity vector.
• Optimal path is a parabolic arc that terminates
  at the vertex of the parabola.

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Fixed Terminal Heading Time-Optimal Trajectory

• Time-Optimal trajectory for an acute-heading
  angle change

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Fixed Terminal Heading Time-Optimal Paths

• Time-Optimal paths for various terminal headings

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Fixed Terminal Position, Free Terminal Velocity

• Terminal Conditions
• Transversality Condition
• Optimal paths are parabolic arcs described by the
  parametric equations
• Taking cross product with and
  gives,
• Letting
  , we get a quartic equation in
• which can be solved to get upto four values for
  . The one that yields
• the minimum value for is the optimal
  solution.

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Fixed Terminal Position Time-Optimal Paths

• Time-Optimal paths for various terminal positions

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Mixed Constraint

• The task at hand is to steer the vehicle to a
  point xf with velocity vf
• in the minimum possible time tf such that the
  vehicle now points
• directly towards the point xa.

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Mixed constraint

• Terminal conditions
• Transversality conditions
• We consider the case when the optimal trajectory
  does not
• terminate in rest. Then,
• optimal acceleration time history takes a
  constant value
• It is directed perpendicular to the final
  velocity vector.
• The optimal path is thus a parabolic arc.

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Mixed Constraint Optimal Acceleration

• The instantaneous position and velocity of the
  vehicle are give by,
• Pre-multiplying by at
  gives,
• Substituting the expression for the final time
  gives,
• letting
  yields a quartic expression in
• which can be solved to obtain up to four values
  for the orientation .
• Among these, the one that yields the minimum
  value for the terminal time is
• the optimal solution.

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Mixed Constraint Time-Optimal Paths

• Time-Optimal paths for various target points

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Length-Optimal Paths

• Consider the problem of a vehicle moving in a
  2-D plane
• from a given initial point and velocity to a
  specified terminal
• heading along the shortest possible path under a
  magnitude
• constraint on acceleration.
• Kinematics
• Problem Statement Find the time history for
  such that
• the system is transferred from to a
  specified terminal condition.
• the length of the path traveled
  is a minimum.
• 
  

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Necessary Conditions for Length-Optimality

• Hamiltonian
• Necessary Conditions

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Transversality Conditions for Length-Optimality

• Terminal conditions
• Transversality conditions
• Substituting in the equation for
  the Hamiltonian
• yields,
• 
  
  
• 
  
  
• 
  
  

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Special Property of the Length-Optimal Paths

• Proposition The instantaneous angle bisector
  between the
• velocity vector and the acceleration vector is a
  constant
• along every optimal trajectory.
• Proof Let be given by,
• is a constant along the optimal
  path.

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Length-Optimal Acceleration Time History

• Instantaneous direction of the optimal
  acceleration vector can be obtained by
• reflecting the instantaneous direction of the
  velocity vector about the fixed
• direction of the angular bisector.
• Let denotes the
  unit vector along the angle bisector and
  denote
• the orthogonal matrix representing reflection
  about given by
• 
  
  
• Therefore the optimal acceleration profile for
  achieving the shortest path that yields a
• desired terminal heading is given by the above
  expression.
• Terminal conditions hold if the optimal
  trajectory
• terminates in rest
• does not terminate in rest
  .

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Length-Optimal Trajectories

• Optimal trajectory terminating in rest
• Trajectory ends in rest only if the heading
  change is obtuse.
• Optimal acceleration vector is opposite to the
  initial velocity vector.
• The time-optimal paths and the length-optimal
  paths are the same which are straight lines.
• Optimal trajectory not terminating in rest
• Trajectory does not end in rest only if the
  required heading change is between zero to 45
  degrees.
• The optimal acceleration vector is obtained by
  reflecting the velocity vector about the fixed
  direction of the angle bisector. .

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Length-Optimal Paths

• Length-Optimal paths for various terminal headings

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Terminal Time and Terminal Arc Length for
Time-Optimality
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Terminal Time and Terminal Arc Length for Length
Optimality
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Comparison of Plots

• Time and Length optimal paths
  Velocity magnitude with time
• Terminal time with heading angle
  Terminal arc length with heading angle

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Conclusion

• Time-Optimal paths for each of the terminal
  constraints considered, are either straight
  lines or parabolic arcs.
• Length-Optimal paths possess a special property
  whereby the angle bisector between the
  acceleration and velocity vectors is a constant.
• Unanswered Questions
• The Time-Optimal path for the mixed terminal
  constraint when the optimal trajectory terminates
  in rest.
• Geometry of the length-optimal paths.
• The length-optimal path for heading changes
  between 45 to 90 degrees.

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FUTURE WORK

• The geometry of the length-optimal paths.
• The time-optimal and length-optimal problems need
  to be solved for achieving a specified terminal
  position and heading.

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THANK YOU