Roboticsaima.eecs.berkeley.edu/slides-pdf/chapter25.pdf · Robotics Chapter 25 Chapter 25 1....
Transcript of Roboticsaima.eecs.berkeley.edu/slides-pdf/chapter25.pdf · Robotics Chapter 25 Chapter 25 1....
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Robotics
Chapter 25
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Outline
Robots, Effectors, and Sensors
Localization and Mapping
Motion Planning
Motor Control
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Mobile Robots
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Manipulators
R
RRP
R R
Configuration of robot specified by 6 numbers⇒ 6 degrees of freedom (DOF)
6 is the minimum number required to position end-effector arbitrarily.For dynamical systems, add velocity for each DOF.
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Non-holonomic robots
θ
(x, y)
A car has more DOF (3) than controls (2), so is non-holonomic;cannot generally transition between two infinitesimally close configurations
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Sensors
Range finders: sonar (land, underwater), laser range finder, radar (aircraft),tactile sensors, GPS
Imaging sensors: cameras (visual, infrared)Proprioceptive sensors: shaft decoders (joints, wheels), inertial sensors,force sensors, torque sensors
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Localization—Where Am I?
Compute current location and orientation (pose) given observations:
Xt+1Xt
At−2 At−1 At
Zt−1
Xt−1
Zt Zt+1
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Localization contd.
xi, yi
vt ∆t
t ∆t
t+1
xt+1
h(xt)
xt
θt
θ
ω
Z1 Z2 Z3 Z4
Assume Gaussian noise in motion prediction, sensor range measurements
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Localization contd.
Can use particle filtering to produce approximate position estimate
Robot positionRobot position
Robot position
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Localization contd.
Can also use extended Kalman filter for simple cases:
robot
landmark
Assumes that landmarks are identifiable—otherwise, posterior is multimodal
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Mapping
Localization: given map and observed landmarks, update pose distribution
Mapping: given pose and observed landmarks, update map distribution
SLAM: given observed landmarks, update pose and map distribution
Probabilistic formulation of SLAM:add landmark locations L1, . . . , Lk to the state vector,proceed as for localization
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Mapping contd.
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3D Mapping example
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Motion Planning
Idea: plan in configuration space defined by the robot’s DOFs
conf-3
conf-1conf-2
conf-3
conf-2
conf-1
w
w
elb
shou
Solution is a point trajectory in free C-space
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Configuration space planning
Basic problem: ∞d states! Convert to finite state space.
Cell decomposition:divide up space into simple cells,each of which can be traversed “easily” (e.g., convex)
Skeletonization:identify finite number of easily connected points/linesthat form a graph such that any two points are connectedby a path on the graph
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Cell decomposition example
start
goal
startgoal
Problem: may be no path in pure freespace cellsSolution: recursive decomposition of mixed (free+obstacle) cells
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Skeletonization: Voronoi diagram
Voronoi diagram: locus of points equidistant from obstacles
Problem: doesn’t scale well to higher dimensions
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Skeletonization: Probabilistic Roadmap
A probabilistic roadmap is generated by generating random points in C-spaceand keeping those in freespace; create graph by joining pairs by straight lines
Problem: need to generate enough points to ensure that every start/goalpair is connected through the graph
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Motor control
Can view the motor control problem as a search problemin the dynamic rather than kinematic state space:
– state space defined by x1, x2, . . . , x1, x2, . . .
– continuous, high-dimensional (Sarcos humanoid: 162 dimensions)
Deterministic control: many problems are exactly solvableesp. if linear, low-dimensional, exactly known, observable
Simple regulatory control laws are effective for specified motions
Stochastic optimal control: very few problems exactly solvable⇒ approximate/adaptive methods
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Biological motor control
Motor control systems are characterized by massive redundancy
Infinitely many trajectories achieve any given task
E.g., 3-link arm moving in plane throwing at a targetsimple 12-parameter controller, one degree of freedom at target11-dimensional continuous space of optimal controllers
Idea: if the arm is noisy, only “one” optimal policy minimizes error at target
I.e., noise-tolerance might explain actual motor behaviour
Harris & Wolpert (Nature, 1998): signal-dependent noiseexplains eye saccade velocity profile perfectly
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Setup
Suppose a controller has “intended” control parameters θ0
which are corrupted by noise, giving θ drawn from Pθ0
Output (e.g., distance from target) y = F (θ);y
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Simple learning algorithm: Stochastic gradient
Minimize Eθ[y2] by gradient descent:
∇θ0Eθ[y
2] = ∇θ0
∫Pθ0
(θ)F (θ)2dθ
=∫ ∇θ0
Pθ0(θ)
Pθ0(θ)
F (θ)2Pθ0(θ)dθ
= Eθ[∇θ0
Pθ0(θ)
Pθ0(θ)
y2]
Given samples (θj, yj), j = 1, . . . , N , we have
ˆ∇θ0Eθ[y2] =
1
N
N∑j=1
∇θ0Pθ0
(θj)
Pθ0(θj)
y2
j
For Gaussian noise with covariance Σ, i.e., Pθ0(θ) = N(θ0, Σ), we obtain
ˆ∇θ0Eθ[y2] =
1
N
N∑j=1
Σ−1(θj − θ0)y2
j
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What the algorithm is doing
x
xx
x x
xx
x xx
xx
xx x x
xxxx
xx
x
xxxx
xxx
x
xx
x
xx
x
x xx
xxx
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Results for 2–D controller
4
5
6
7
8
9
10
0.2 0.4 0.6 0.8 1 1.2
Vel
ocity
v
Angle phi
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Results for 2–D controller
4.51
4.52
4.53
4.54
4.55
4.56
4.57
4.58
4.59
4.6
4.61
0.6 0.61 0.62 0.63 0.64 0.65
Vel
ocity
v
Angle phi
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Results for 2–D controller
0.0055
0.006
0.0065
0.007
0.0075
0.008
0.0085
0.009
0.0095
0 2000 4000 6000 8000 10000
E(y
^2)
Step
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Summary
The rubber hits the road
Mobile robots and manipulators
Degrees of freedom to define robot configuration
Localization and mapping as probabilistic inference problems(require good sensor and motion models)
Motion planning in configuration spacerequires some method for finitization
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