application of geometrical extrapolation method based hybrid system controller on pursuit-avoidance...
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APPLICATION OF GEOMETRICAL APPLICATION OF GEOMETRICAL EXTRAPOLATION METHOD BASED EXTRAPOLATION METHOD BASED
HYBRID SYSTEM CONTROLLER ON HYBRID SYSTEM CONTROLLER ON PURSUIT-AVOIDANCE DIFFERENTIAL PURSUIT-AVOIDANCE DIFFERENTIAL
GAMEGAME
Ginzburg Pavel and Slavnaya Lyudmila
Supervisory by Dr. Mark Moulin
List of contents:
Optimal control and Hamiltonian formalism Hybrid control Intuitive Geometrical Controller Circle Extrapolation Controller Second Order Approximation in Geometrical
Extrapolation Nonlinear Heading Estimator The Supper Hybrid Controller Conclusion
Hamiltonian Formalism
System representation
Solutions of Backward Reachable Set
Capture radius 5Linear velocities 5
Angular velocities 1
Capture radius 5Linear velocities 5
Angular velocity Evader 2Angular velocity Pursuer 1
Capture radius 5Linear velocities 5
Angular velocity Evader 1Angular velocity Pursuer 2
Open loop via closed loop control
Open loop:+ Safe for initial condition check - Sensitive to measurement noise - Non flexible to task changing - Slow and massive calculation (offline ) - And more…
Closed loop: + Safe control + Online and fast calculations - Non optimal
Intuitive Geometrical Controller
Plane division to provide control signal
Problematic situation for Simple Controller
Results of Intuitive Geometrical Controller
Initial conditions (-2, 2, π/2)
Initial conditions (-2, 2, π/3) Control signal output
Control signal output
Circle Extrapolation Controller
Red Cross – estimated positionBlue Star – measurement data, stored in controller memory
Assumptions:
- The opponent control signal unchanged during the sampling
Principe:
- 3 points define the only one circle
- Forth point is the opponent estimated position
-Controller chooses the optimal output depends on the estimated position
- Each step correction provided (like Kalman filter measurement correction)
Results of Circle Extrapolation Controller
Initial conditions (-2, 2, π/2) Control signal output
Initial conditions (-2, 2, π/3) Control signal output
Second Order Approximation in Geometrical Extrapolation
Results Geometrical Extrapolation Controller
Initial conditions (-2, 2, π/3)
Initial conditions (-2, 2, π/2) Control signal output
Control signal output
Geometrical Extrapolation Controller used by both players
Border point (0, 3.44, -π)Applied control signals
Applied control signalsBorder point (2, 1.42, -π/2)
(0, 3, -π)
(1.6, 1, -π/2)
Results Geometrical Extrapolation Controller with “arctan”
“Sign” function replaced by “Arctangent”
Initial conditions (-2, 2, π/2) Control signal output
Control signal output Initial conditions (-2, 2, π/3)
Noise in coordinate measurement
1 variance noiseBorder point (2, 1.42, -π/2)
0.1 variance noiseBorder point (2, 1.42, -π/2)
Nonlinear Heading Estimator Assumptions: - The opponent heading signal unchanged during the sampling Principe: - 5 points is a trade off between noise filtering and device flexibilityController provides the most fitted line (MSE)
Red points – measurement dataBlue lines – calculated slopes, the estimator output
Performances of Nonlinear Heading Estimator
Constant control input (0), noisy case (0.01 variance)
no noise exist
Constant control input (0.1), noisy case (0.01 variance).
The Supper Hybrid Controller
Idea:- fitting noisy data to estimated orbit - verify the noise level by R-condition checking - choose more successful controller
implementation
Travels between hybrids controllers (double hybridism exists)
Conclusions
Advantages of Hybrid Control Implementation The geometrical extrapolation method in
Hybrid system provides wise plane division Noise filtering using The Method More levels in Hybrid Implementation
Directions
Non linear filtering (fit the surface to measurement data)
Analytical approximated solution of HJB equations
More complex differential games General case