Parameter curve notes of coursera self driving car Part4 motion planning

         As Coursera Chapter four FinalProject Theoretical preparation for , Tidy up Coursera Driverless courses 4.7.1 parametric curves The notes .

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Catalog

Learning goals

Kinematic constraints

Parametric curve  

Path optimization

Path parameterization example

  Quintic spline

Quintic spline curvature calculation

Cubic helix

Polynomial helix x,y Coordinate calculation

summary

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Learning goals

1. Understand the path planning problem and its constraints and boundary conditions
2. Know what the parametric curve is
3. Describe the advantages and disadvantages of using spline and helix in path planning

Kinematic constraints

Maximum curvature limit ： Because the vehicle has a minimum turning radius , It means that the curvature of the path that the vehicle can walk cannot exceed the maximum value .

Usually for the sake of simple calculation , By limiting the curvature of several points on the curve, the whole curve is constrained .

Parametric curve  

Original y=y(x) The curve of is written as x(u),y(u) In the form of ,x,y The value of is determined by the parameter u To make sure , adopt u To traverse the entire curve .

Parameters u Can be arc length , It can also be dimensionless [0,1] 0 Represents the starting point ,1 It's for the end .

Path optimization

Through the cost function f To optimize the path , Parametric curves allow optimization in the parameter space , Can simplify optimized expressions .

They are constraints c(r(u))≤α And boundary conditions r(0)=β0, r(uf)=βf

Path parameterization example

Two commonly used parametric curves are quintic polynomial and cubic helix

  Quintic spline

x,y All are u Of 5 Sub polynomial , There is an analytical solution for the boundary condition (closed form solution)

Quintic spline curvature calculation

  For quintic splines , The challenge is the curvature constraint , Because of its potential curvature discontinuity , The denominator may be 0 And the curvature derivative may be 0 You can't 1 Order continuity . Curvature calculation is complicated , It is difficult to calculate constraints .

Cubic helix

Helix defines curve curvature as a function of arc length , Definition makes curvature constraint checking simple , Curvature can be constrained by simply sampling some points on the curve to obtain good curvature performance , Because of the properties of polynomials .

k(s) Is the curvature equation

θ(s) It can be understood as the increased heading angle relative to the starting point delta heading.

Polynomial helix x,y Coordinate calculation

There is no exact analytical solution for the position of the helix , Only by numerical methods - Simpson's law to approximate Fresnel integral .

summary

The boundary conditions and constraints in path planning are discussed

The parameter curve is introduced

The difference between spline and helix in path planning is discussed