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COS 495 - Lecture 2 Autonomous Robot Navigation
Instructor: Chris Clark Semester: Fall 2011
Figures courtesy of Siegwart & Nourbakhsh
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Navigation and Control
1. Control Architectures 2. Navigation Example 3. Basic Tools for AUV Navigation
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Control Architectures
§ Today, most robots control systems have a mixture of planning and behavior-based control strategies.
§ To implement these strategies, a control architecture is used.
§ Control architectures should be: § Modular § Localized
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Control Architectures Desired Characteristics
§ Code Modularity § Allows programmers to interchange
environment types sensors, path planners, propulsion, etc.
§ Localization § Embed specific navigation functions within
modules to allow different levels of control (e.g. from task planning to wheel velocity control)
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Control Architectures Decomposition
§ Decomposition allows us to modularize our control system based on different axes:
1. Temporal Decomposition – Facilitates varying degrees of real-time processes
2. Control Decomposition – Defines how modules should interact: serial or
parallel?
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Control Architectures Temporal Decomposition
§ Factors affecting temporal decomposition: § Sensor response time § Temporal memory and
horizon § Spatial Locality § Context Specificity
T. Memory & Horizon
Context Specificity
Spatial Locality
Sensor Response Time
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Control Architectures Temporal Decomposition
§ Example
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Control Architectures Tiered Architectures
§ A general tiered architecture for episodic planning § Role of the Executive is:
§ Switch behaviors § Monitor failures § Call the planner
§ Planning only when required (e.g. blockage)
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Control Architectures Tiered Architectures
§ A tiered architecture for integrated planning
§ Planning is fast and is embedded as a behavior.
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Control Architectures Control Decomposition
§ An example of a control decomposition using a mixture of serial and parallel approaches.
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Navigation and Control
1. Control Architectures 2. Navigation Example
1. Motion Modeling 2. Estimation and Control 3. Experiments
3. Basic Tools for AUV Navigation
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Navigation Example
§ Autonomous Control § Goal is to enable autonomous trajectory tracking
capabilities. § Given individual ROVs have autonomous control,
multi-vehicle control research will be facilitated.
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Equations of Motion
§ 6 degrees of freedom (DOF): § State vectors:
body-fixed velocity vector: earth-fixed pos. vector:
DOF Surge Sway Heave Roll Pitch Yaw
Velocities u v w p q r
Position & Attitude x y z φ θ ψ
Forces & Moments X Y Z K M N
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§ The 6-DOF nonlinear dynamic equations of motion can be expressed as: where: inertia matrix: Coriolis & centripetal matrix: hydrodynamic damping: restoring forces: propulsion forces:
νηη
τηννννν
)()()()(
JgDCM
=
=+++
)()()( ννν ARB CCC +=ARB MMM +=
)(νD)(ηg
τ
Equations of Motion
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Equations of Motion
§ Initial Assumptions: § The ROV will usually move with low velocity when on mission § Almost three planes of symmetry; § Vehicle is assumed to be performing non-coupled motions. § Horizontal Plane:
§ Vertical Plan: �
[W. Wang et al., 2006]
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Theory vs. Experiment
§ Coefficients for the dynamic model are pre-calculated using strip theory;
§ A series of tests are carried out to validate the hydrodynamic coefficients, including § Propeller mapping § Added mass coefficients § Damping coefficients
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Propeller Thrust Mapping
§ The forward thrust can be represented as:
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Direct Drag Forces
Drag in Heave Direction
Drag in Sway Direction
Drag in Surge Direction
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Estimation & Control
§ Sensor Overview
§ VideoRay Compass § VideoRay Depth
Sensor § Desert Star Acoustic
Positioning System
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Estimation & Control
§ State Estimation § We fuse several sensor measurements using an Unscented Kalman Filter:
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Estimation & Control § Trajectory Tracking
§ We use three different controllers:
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Estimation & Control
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Estimation & Control
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Autonomous Control
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Autonomous Control
§ Hardware Modifications q Added transceiver q Added bouyancy q Shifted weight q Extended feet
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Autonomous Control
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Autonomous Control
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Autonomous Control
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Swimming Pool Experiments
§ Sample run: x, y state estimates
-1 0 1 2 3 4 5 6 7 8-1
0
1
2
3
4
5
6
7
8
estimatemeasurementreference
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Swimming Pool Experiments
§ Sample run: bearing state estimates
0 10 20 30 40 50 60 700
1
2
3
4
5
6
7
time (seconds)
bear
ing
(radi
ans)
estimatemeasurement
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Swimming Pool Experiments
§ Sample run: depth state estimates
0 10 20 30 40 50 60 700.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
time (seconds)
dept
h (m
eter
s)
estimatemeasurementreference
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Swimming Pool Experiments
Trial # Mean of x (m) Standard Deviation (m) 1 3.186 0.431 2 2.906 0.495 3 3.095 0.129 4 3.040 0.137 5 3.192 0.179 6 2.890 0.265 7 2.966 0.154
[W. Wang et al., 2006]
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Navigation and Control
1. Control Architectures 2. Navigation Example 2 3. Basic Tools for AUV Navigation
1. Coordinate Frames 2. Motion Modeling 3. P Control
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Cartesian Coordinates
§ Describes unique position of points in a plane with respect to the axis
§ For each dimension there is 1 axis
§ Coordinates are measured in “units” in the direction parallel to the axis
§ The origin is fixed to the plane
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Cartesian Coordinates
§ One can use cartesian coordinates to describe a robot’s position
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Cartesian Coordinates
§ For our underwater robots, we need 3 degrees of freedom to express position
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Polar Coordinates
§ In polar coordinates, we specify points on a 2D plane using the length of a radius arm and an angle
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Cylindrical Coordinates
§ For specifying point locations in 3D, cylindrical coordinates can be used by specifying the length of a radius arm, an angle, and a height.
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Polar to Cartesian
§ How do we convert from polar coordinates to Cartesian coordinates?
x = r cos(θ) y = r sin(θ)
y
x
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Global (Inertial) Coordinate frame
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Global (Inertial) Coordinate frame
§ Anchor a coordinate frame to the environment
XI
YI
θ
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Global (Inertial) Coordinate frame
§ With this coordinate frame, we describe the robot state as: XI = [x y θ]I
XI
YI
θ
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Local Coordinate frame
§ Anchor a coordinate frame to the robot
XR YR
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Local Coordinate frame
§ With this coordinate frame, we describe the robot state as: XR = [x y θ]R = [0 0 0]
XR YR
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Local Coordinate frame
§ The local frame is useful when considering taking measurements of environment objects. § Consider the detection of an wall using a range
finder:
XR YR
ρobject
αobject object
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Local Coordinate frame
§ The measurement is taken relative to the robot’s local coordinate frame (ρobject, αobject)
§ We can calculate the position of the measurement in local coordinate frames: xobject, R = ρobject cos( αobject ) yobject, R = ρobject sin( αobject )
XR YR
ρobject
αobject object
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Local Coordinate frame
§ One can calculate the position of the object in the global coordinate frame ( xobject,I , yobject,I )
xobject, I = xAUV, I +( xobject,R cos (θAUV, I ) ) + ( yobject,R sin (θAUV, I ) )
yobject, I = yAUV, I +( xobject,R sin (θAUV, I ) ) + ( yobject,R cos (θAUV, I ) )
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Local Coordinate frame
§ One can calculate αobject if the positions are known
θAUV, I + αobject = atan2( yobject – yAUV, xobject – xAUV )
αobject
object
θAUV
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Navigation and Control
1. Control Architectures 2. Navigation Example 2 3. Basic Tools for AUV Navigation
1. Coordinate Frames 2. Motion Modeling 3. P Control
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A (simple) motion model
§ Consider a robot moving from position x, y in direction θ radians with forward velocity ν m/s and rotational velocity w rad/s.
θ
ν
x,y
w
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A (simple) motion model
§ How much will it rotate in t seconds? It will rotate a distance of wt radians.
θ w
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A (simple) motion model
§ So, from time step k to time step k+1, the angle changes to:
θk+1 = θk + wt
θ w
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A (simple) motion model
§ So, from time step k to time step k+1, the position changes to:
xk+1 = xk + vt cos( θk ) yk+1 = yk + vt sin( θk )
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Navigation and Control
1. Control Architectures 2. Navigation Example 2 3. Basic Tools for AUV Navigation
1. Coordinate Frames 2. Motion Modeling 3. P Control
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P Control
§ Proportional Feedback Control – P Control – uses the error between the desired and measured state to determine the control signal.
§ If xdesired is the desired state, and x is the actual state, we define the error as: e = xdesired – x
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P Control
§ The control signal u is calculated as
u = KP e
where KP is called the proportional gain.
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P Control
§ Example: § Consider the orientation control of an AUV. Assume
the orientation is completely controlled by the rear rudder fins.
u
θ
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P Control
§ Example cont’: § The control signal u is calculated as
u = KP(θdesired - θ) § Notes:
§ If θdesired = θ, the control signal is 0. § If θdesired < θ, the control signal is negative, resulting in an
decrease in θ. § If θdesired > θ, the control signal is positive, resulting in an
increase in θ. § The magnitude of the increase/decrease depends on Kp
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P Control
§ Block Diagram: u = KP(θdesired - θ)
KP AUV θ u e
- + θdesired
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P Control
§ Time Domain Response of step response § Step from θdesired = 0 to θdesired = 1.
θdesired
time
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P Control
§ Time Domain Response: § Step from θdesired = 0 to θdesired = 8. § Different dynamics in this example… overshoot!
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Lab
§ Form a group of Three § Email instructor one list of names along with a group
number
§ Start Lab 0 § Optional for those with MVS and C# experience
§ Start Lab 1 § Code can be downloaded from the internet
§ Read Lab 3 § Can brainstorm in your groups