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Introduction to Dynamical Systems Theory


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describe a simulated six-legged robot
legs swing about a single joint with the body
five neurons to control a single leg
describe the network architecture of the locomotion controller
50 (network weights)
how many parameters consist of a neural controller?
via binary string
how is the neural controller encoded?
what is the fitness function of a neural controller based on?
it's bsaed on the distance traveled in a fixed time period
the TRIPOD gait
what is the gait favored for fast walking?
via dynamical systems (no need to memorize equations)
how can we quantitatively evaluate the types of tripod walkers?
what the basic components of a dynamical system?
system, state, state space, state variables, trajectory, phase portrait, attractors, basin, repellors, saddle
define a system
a set of interacting factors (called state variables) whose values change over time (population of owls, mice, and availability of grain)
define a state
vector of values, one for each variable of the system at a given moment (20 owls, 30 mice, 1.5 lb of grain)
define a state space
all possible states of the system (# of owls, mice, amt of grain left)
define an instantaneous velocity vector
rate/direction of change in teh state of the system at a point in time
vector field
collection of all of the instantaneous velocity vectors--TECHNICALLY, a dynamical system is equivalent to this vector field
define a trajectory
cruve connecting temporally successive points in a state space
define a phase portrait
a state space filled with trajectories of a given model
the trajectories (phase portrait)
what gives the history of change of the system over time
describe a periodic trajectory
it is a cyclic, closed trajectory--it closes upon itself
describe a fixed point in a phase portrait
a point in the state space with zero instantaneous velocity (constant, critical, rest point)
desribe an attractor
limit sets to which all nearby trajectories tend towards--can be a fixed attractor, periodic, or chaotic
define a basin
region of the state space containing all trajectories which tend to a given attractor
define a repellor
points and periodic trajectories from which trajectories only leave
define a saddle
limit sets which some trajectories approach and others depart
define a chaotic trajectory
trajectories that are neither fixed nor cyclic but which fill up a constrained region of the state space (does not go to a fixed point or a cycle, but remains constrained in a region of phase space)
using a 2-neuron system
how can we produce such dynamical systems?
define a Lorenz attractor
used to study weather phenomena, and traces a single trajectory
define a strange attractor
all particles converge to a same volume
agent and its environment as coupled dynamical systems
how can we use dynamical systems to model robotics and evolution?
define the three classes of evolved controllers
if sensors were enabled during evolution, then reflexive pattern generators evolved
define the 5D projection of the pattern generators/evolved controllers
it's a 5D limit cycle, motor space projection, plots the output of foot, backward swing, forward swing
using the 5D projections; compare 3D curves
how can we compare behavior between evolved controllers?