Controller Synthesis
in Complex Environments
Nora Ayanian
March 20, 2006
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Introduction
• Many different approaches to robot motion planning
and control
 Continuous: Navigation function
• Configuration space must be a generalized sphere world
• Any vehicle dynamics
 Combined continuous and discrete: Decomposition of state
space
• Can handle more complex configuration space
• Difficulty with complex dynamics
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Continuous Method
• Rimon and Koditschek [1] present a method to guide a
bounded torque robot to a goal configuration from
almost any initial configuration in an environment that
is:




Completely known
Static
Deformable to a sphere world
Admits a navigation function
• Create an artificial potential field that solves the three
separate steps of robot navigation
 Path planning
 Trajectory planning
 Control
[1] *E. Rimon and D.E. Koditschek, “Exact Robot Navigation Using Artificial Potential Functions,” IEEE Transactions on
Robotics and Automation, vol. 8, no. 5, pp. 501-518, 1992.
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Continuous Method
• Let V be a map
 With a unique minimum at the goal configuration, qd
 That is uniformly maximal over the boundary of the free
space, F
• V determines a feedback control law of the form
 ( p, p )  V ( p)  d ( p, p )
• The robot copies the qualitative
behavior of V’s gradient [2]
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Navigation Function Method
• Star shaped sets
 Star shaped sets contain a distinguished “center point” from
which all rays cross the boundary of the set only once.
 Map the star onto a disk diffeomorphically: translated scaling
map
Ti (q)  n i  [q  qi ]  pi

n i (q)  [1   i (q)]1 2
q  qi
 Scales each ray starting at qi by ni, then translates along pi
S
D
qi
pi
*[1] Rimon & Koditschek
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Combined Continuous and Discrete Method
• Habets & van Schuppen [7] decompose the state
space into polytopes
• Each polytope is a different discrete mode of the
system
• Objective: steer the state of an affine system to a
specific facet
• Focus is on simplices
 Points contained in a simplex are
described by a unique linear
combination of the vertices
[7] *L.C.G.J.M. Habets and J.H. van Schuppen, “A Control Problem for Affine Dynamical Systems on a Full-Dimensional
Polytope,” Automatica, no. 40, pp. 21–35, 2004.
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Combined Method: Problem Definition
x  Ax  Bu  a, x(0)  x0 on PN
• Consider the affine system
• For any initial state x0  PN, find a time instant T0 ≥ 0 and
an input function u: [0,T0]  U, such that
 t  [0,T0]: x(t)  PN ,
 x(T0)  Fj, and T0 is the smallest time-instant in the interval [0,∞)
for which the state reaches the exit facet Fj
 nTj x (T0 )  0 , i.e. the velocity vector at the point x(T0)  Fj has a
positive component in the direction of nj. This implies that in
the point x(T0), the velocity vector points out of the polytope PN.
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Combined Method: Necessary Conditions
• If the control problem is solvable by a continuous state
feedback f, then there exist inputs u1,…,uM  U such that
  j  V1:
• n1T(Avj + Buj + a) > 0,
•  i  Wj \ {1}: niT(Avj + Buj + a) ≤ 0.
  j  {1,…,M} \ V1:
•  i  Wj : n1T(Avj + Buj + a) ≤ 0,
•

i
Illustration of
Polyhedral Cones
niT ( Av j  Bu j  a)  0
Habets & van Schuppen,2004
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Applying the Combined Method
• A 1-dimensional integrator problem
 x  0 1  x  0
 x  0 0  x   1u
  
   
x'
x
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Thank You
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