introduction to robotics - cse - iit kanpur · introduction to robotics amitabha mukerjee iit...
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Introduction to Robotics
Amitabha Mukerjee
IIT Kanpur, India
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What is a Robot? Mobile Robots
Robot properties:
Flexibility in Motion
Mobile robots
daksh ROV: de-mining robot 20 commissioned in Indian
army 2011. 100+ more on order
built by R&D Engineers, Pune
daksh platform derived gun mounted robot (GMR)
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What is a Robot? Articulated Robots
Robot properties:
Flexibility in Motion
Mobile robots
Articulated Robots
Soccer playing humanoid robot[http://labintsis.com]
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Robot you can own
Roomba vacuumCleaning robot
By i-robotPrice: ~ rs. 15-30K
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Algorithms for Robot motion
Roomba vacuumCleaning robot
By i-robotPrice: ~ rs. 30K https://www.youtube.com/watch?v=dweVBqei9L
A
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Models of Robot Motion
Circular robot
World Frame(Workspace frame)
W
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Models of Robot Motion
Robot frame
R
y
x
y
xWorld Frame(Workspace frame)
DEFINITION: degrees of freedom:
number of parameters needed to fix the robot frame Rin the world frame W
NOTE: Given robot frame R, every point on the robot is known
W
given configuration qfor a certain pose of the robot, the set of points on the robot is a function of the configuration: say R(q)
(x,y) = configuration(vector q)
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Non-Circular Robot
DEFINITION: degrees of freedom:
number of parameters needed to fix the robot frame Rin the world frame W
How many parameters needed to fix the robot frame if it can only translate?
How many if it can rotate as well?
W
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Motion in 3-D: Piano movers problem
General 3D motion:
How many parameters needed to fix the pose?
Can a design be assembled?
Test based on CAD models
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Research mobile robot
Turtlebot
Based on i-robot (roomba) platform(with kinect RGB-D sensor)
ROS (open-source) software
Price: ~ 75K
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Articulated robots
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What is a Robot?
Robots properties:
Flexibility in Motion
Mobile robots
Articulated robots
SCARA 4-axis arm(4 degrees-of-freedom)
by Systemantics Bangalore
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Industrial Robot
Robots involve
Flexibility in Motion
Mobile robots
Articulated robots
Industrial robot
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Industrial Robots
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How to program a welding robot?
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What is a Robot?
Robot properties
Flexibility in Motion
Mobile robots
Articulated robots
Industrial robot
Surgicalrobots
perfint healthcare, chennai
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Surgical Robot : Lumbar biopsy
needle path as planned on CAT scan
inserted needle position
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Modeling Articulated Robots
Kinematic chain: Pose of Link n depends on the poses of Links 1...(n-1)
Transformation between frame of link (n-1) and link n, depends on a single motion parameter, say θn
Exercise:What are the coordinates of the orgin of the end-effector center?
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Modeling Articulated Robots
workspace configuration space
θ1
θ2
Exercise:Sketch the robot pose for the configuration [0, -90]
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Modeling Articulated Robots
Forward kinematicsMapping from configuration q to robot pose, i.e. R(q)
Usually, R() is the product of a sequence of transformations from frame i to frame i+1.
Note: Must be very systematic in how frames are attached to each link
Inverse kinematicsa. Given robot pose, find qOr b. Given end-effector pose, find q
Q. Is the answer in (b) unique?
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What is a Robot?
Robots properties
Flexibility in Motion
Mobile robots
Articulated robots
Digital actors
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Reality: limited functionality
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Mobility isnt everything
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What is a Robot?
Robots involve
Flexibility in Motion
Dentists cradle?
Washing machine?
Intentionality
Measure : not default probability distribution
e.g. Turn-taking (contingent behaviour)
Goal : intrinsic or extrinsic
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Robot Motion Planning
Amitabha Mukerjee
IIT Kanpur, India
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indian edition rs 425
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Sensing and Motion Planning
Bohori/Venkatesh/Singh/Mukerjee:2005[bohori venkatesh singh mukerjee 05]
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Programming a robot
Aldebaran Nao
Grasping an offered ball
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Programming a robot
1. detect ball using colour:
image captured by nao HSV binarized contour detected
2. estimate distance of ball (depth)from image size
3. Inverse kinematics to grasp ballSensing in the workspace
Motion planning in C-space
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Configuration Space
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Robot Motion Planning
(xS,yS)goal
start
(xG,yG)
Obstacle B
Valid paths will lieamong those where the robot does not hit the obstacle
find path P from start to goal s.t.
for all t, R(t) ∩ B = Ø
How to characterize the set of poses for which the robot does not hit the obstacle B?
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Robot Motion Planning
How to characterize the set of poses for which the robot does not hit the obstacle B?
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Continuum approaches vs Discretization
Two approaches to Robot motion planning:
• continuum: treat motion space as single continuum
optimization
• discretization:decompose motion space into regions / segments
graph-search
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Potential fields
Potential fields
1. Goal: negative (attractive) potential Obstacles: positive (repulsive) potential
2. Robot moves along gradient
3. Problems:
- need to integrate the potential over the area of robot
- problem of local minima
goal
start
Obstacle B
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Finite area robots
Instead of integrating over robot area, restrict to a set of control pointse.g. vertices
Problem: With control points r1 and r2 on robot R(q), edge E1 may still hit Obstacle.
Attempt to reduce computation to points
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Local Minima
persists even for point robots
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Nature of Configuration spaces
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Robot Model
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Models of Robot Motion
Robot frame
R
y
x
y
xWorld Frame(Workspace frame)
DEFINITION: degrees of freedom:
number of parameters needed to fix the robot frame Rin the world frame W
W
given configuration qfor a certain pose of the robot, the set of points on the robot is a function of the configuration: say R(q)
(x,y) = configuration(vector q)
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Robot Motion Planning
find path P from qS to qG s.t. for all q ϵ P, R(q) ∩ B = Ø
? generate paths and check each point on every path?
Would it be easier to identify Qfreefirst?
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Robot Motion Planning
Q QB
QB = [ q | R(q) ∩ B ≠ Ø }
goal qstart q
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Motion Planning in C-space
Q
start q
pathconfigurations are points in C-space
path P is a line
if P ∩ QB = Ø, then path isin Qfree
QB
goal q
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Motion Planning in C-space
Qworkspace
Configuration space
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startgoal
CB
Robot Motion Planning
workspace W
configuration space C
path
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Non-circular mobile robotsTriangle - translational edges of C-obstacle
are parallel to obstacle and robot edges...
C-obstacle
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Non-circular mobile robots
C-space with rotation θ (polygonal obstacle)
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Configuration Space for Articulated Robots
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Configuration Space Analysis
Basic steps (for ANY constrained motion system):
1. determine degrees of freedom (DOF)
2. assign a set of configuration parameters qe.g. for mobile robots, fix a frame on the robot
3. identify the mapping R : Q →W, i.e. R(q) is the set of points occupied by the robot in configuration q
4. For any q and given obstacle B, can determine if R(q) ∩ B = Ø. can identify Qfree
Main benefit: The search can be done for a point
5. However, computation of C-spaces is not needed in practice; primarily a conceptual tool.
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Articulated Robot C-space
How many parameters needed to fix the robot pose ?
What may be one assignment for the configuration parameters?
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C-space as manifolds
Topology of C-space: Torus (S1 x S1)
Choset, H etal 2007, Principles of robot motion: Theory, algorithms, and implementations, chapter 3
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C-space as manifolds
• manifold: generalization of curves / surfaces
every point on manifold has a neighbourhood homeomorphic to an open set in Rn
• Mapping Φ : S T is bijective (covers all of T and has unique inverse)Φ is homeomorphic:
(f / f-1 are continuous)diffeomorphic :
(f / f-1 are C∞ smooth)
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Neighbourhood of q is mappable to R2
global topology is not R2 but S1 x S1 (torus)
C-space as manifolds
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Map from C-space to W
Given configuration q, determine volume occupied by R(q) in workspace
For multi-link manipulators, spatial pose of link (n+1) depends on joint configuration q for joints 1, 2, ..., n.
Forward Kinematics
Map from W to C-space: given pose in workspace, find q
Inverse Kinematics
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Mapping obstacles
Point obstacle in workspace
Obstacle in Configuration Space
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Articulated Robot C-space
Path in workspace Path in Configuration Space
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Graph-based approaches
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Visibility Graph methods
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Visibility Graph methods
Construct edges between visible vertices
Sufficient to use only supporting and separating tangents
Complexity: Direct visibility test: O(n3)
(tests for each vtx: O(n) emanations x O(n) obst edges)
Plane sweep algorithm: O(n2logn)
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Visibility Graph methods
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Visibility Graph methods
Sufficient to use only supporting and separating tangents
Finds “shortest” path – but too close to obstacles
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Cell decomposition methods
Trapezoidal decomposition: Each cell is convex.
Sweep line construction: O(nlogn)
Graphsearch: O(nlogn)
Path: avoids obstacle boundary but has high curvature bends
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Roadmap methods
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Roadmaps
any roadmap RM must have three properties:
Connectivity: path exists between any q′START and q′GOAL in RM
Accessibility: exists a path from any qSTART ∊ Qfree to some q′START∊ RM
Departability: exists a path from some q′GOAL ∊ RM to any qGOAL ∊Qfree
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Staying away from Obstacles: Generalized Voronoi Graphs
Voronoi Region of obstacle i :
Voronoi diagram:set of q equidistant from at least two obstacles
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Generalized Voronoi Graphs
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GVG Roadmaps
Accessibility / Deparability: Gradient descent on distance from dominant obstacle : guaranteed to reach from any
qSTART ∊ Qfree to some q′START ∊ RM
motion is along a “retract” or brushfire trajectory
Connectivity: GVG is Connected if path exists
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Sensor based Voronoi roadmap construction
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Canny’s Silhouette roadmap
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Canny’s Silhouette roadmap
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Canny’s Complexity Analysis
n: = degrees of freedom of robot (dim of C-space)
obstacles C-space boundaries represented as p polynomials of maximum degree w
Complexity:any navigation path-planning problem can be solved in pn(logp)wO(n4) time
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Probabilistic Roadmap (PRM)
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Probabilistic Roadmap
Sample n poses q1…qn in the WORKSPACE
Free space nodes: Reject qi that intersect with an obstacle, remaining nodes q are in Qfree
Local planning: in k-nearest neighbours, if path <qi,qj>collision-free, add edge to graph
Resulting graph = Probabilistic Roadmap
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Local Planner
Objective: Test if path <qi,qj> is collision-free
Linear Subdivision algorithm: start at midpoint(qi,qj) ; subdivide recursively until desired precision
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Probabilistic Roadmaps (PRM)
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Sampling-based motion planningSample n poses q1…qn in the workspace
Reject qn that overlap with an obstacle, remaining poses are in Qfree
Use local planning to determine if a path exists between neighbours qi and qj.
Resulting graph = Probabilistic Roadmap
Probabilistically complete: As #samples n ∞, Prob (success) 1
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Hyper-redundant robot motion planning using PRM
[sinha mukerjee dasgupta 02]
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Hyper-redundant robot motion planning using PRM
[sinha mukerjee dasgupta 02]
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Hyper-redundant motion planning
[sinha mukerjee dasgupta 02]
Time:Exponential in DOFs
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Design for manipulability
[sinha mukerjee dasgupta 02]
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PRM applications
42 DOFs: [Sánchez and J. C. Latombe 02]
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Narrow corridor problem
Solution: generate more samples near boundary – bias the sample towards boundary region– if midpoint between two obstacle nodes is free, add
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PRM applications : Protein folding
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Sampling based methods: PRM
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Continuum methods: Overcoming Local minima
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Grid-based: Wave-front
• Grid-based model
• given a start grid cell qS assign it the value “2”
• Every neighbour gridcell gets +1
• Until grid is filled
• Given a goal cell qG use greedy search to find path back to goal
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Grid-based: Wave-front
O(kd) space / time
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Navigation Function : Sphere space
[Rimon Koditschek 92]
• Spherical wall (r0), with spherical obstacles inside
• Obstacle distance wallobstacles
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Sphere space
Rimon Koditschek 92
center qiradius ri
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Navigation Function : Sphere space
[Rimon Koditschek 92]
• Spherical wall (r0), with spherical obstacles inside
• Obstacle distance
• Goal potential with high exponent
• Instead of sum, use product to combine obstacle potentials
• For high k, has unique minima at goal
wallobstacles
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Navigation Function
Choset etal 05
k=4
k=8 k=10
k=6
++
+ +
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Navigation Function
Choset etal 05
φ : S → [0, 1] : navigation function on sphere space S.
For any space F if exists diffeomorphicmapping h : F → S(i.e. h is smooth, bijective, and has a smooth inverse),
then φ = φ∘ h is a navigation function on F
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Sensori-motor map learning
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Cognitive Architecture: Levels of AbstractionsEx
tern
al W
orld
Lang
uage
, Log
ic, a
nd C
ogni
tion
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Visuo-Motor expertise
[A. van der Meer, 1997: Keeping the arm in the limelight]
Newborns (10-24 days)
Small weights tied to wrists
Will resist weights to movethe arm they can see
Will let it droop ifthey can't see it
in darkened room,works hard to position arm
in a narrow beam of light
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Observing self motions
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Simulation
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residual variance = 1 – r2(Dg, Dy); r = linear correlation coefficientDg = geodesic distance matrix; Dy = manifold distance
Manifold dimension
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Discovering Configuration Space
Collect a set of images of the robot at random configurations
Reduce the image set to a low dimensional representation
Latent variables: (y1, y2,... yd)for d=2, each image represented as a
pair of values
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Smoothly Deformable objects
Object S = set of connected points S in G ⊂ Rd.
Deformation function h : G G is a function of a parameter vector q = {q1...qk}.
Smooth Deformation : S hS(q), is a diffeomorphism from G to G
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Smoothly Deformable objects
Ex.1
Articulated chain with N rigid links, and k joints, each with 1 DOF each.
hS determined by {q1...qk}, where each qi is the parameter associated with each joint.
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Deformable objects
LEMMA:
The space of shapes and poses of S is a manifold of at most k dimensions. (k = dimensionality of q).
Any x IN S(q) has a neighbourhood N, s.t. there is a homeomorphism phi from N onto the space Q of {q1...qN}. In RN
the map phi can be composed of the motion transformation T(q1..qm) [a special euclidean group) andthe smooth deformation shape funnction hS(q1..qd)
Both of these transformations being diffeomorphic.
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Imaging transformation
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Visual Manifold theorem
If S is a smoothly moving, visually distinguishable, deformable object, then any image of S, taken from a fixed camera, would lie on a manifold of at most N dimensions in the image space.
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Visual Distinguishability
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Robot Structure Discovery
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Simulation
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Robot Structure Learning
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• Latent variables:– Distributed on S1 x S1 topology– Along circumferential path: θ1; along radial: θ2
– Naïve, non-metric representation of θ1,θ2
• Manifold transformation = mapping between input images (workspace) ↔ naive θ1,θ2 (C-space)– manifold → image ≈ naïve forward kinematics– image → manifold ≈ naïve inverse kinematics
Robot Structure Learning
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• Latent variables:– Distributed on S1 x S1 topology– Along circumferential path: θ1; along radial: θ2
– Naïve, non-metric representation of θ1,θ2
• Manifold transformation = mapping between input images (workspace) ↔ naive θ1,θ2 (C-space)– manifold → image ≈ naïve forward kinematics– image → manifold ≈ naïve inverse kinematics
Robot Structure Learning
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Mapping to control parameters
Supervised: Image to (θ1,θ2)
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Mapping to control parameters
Supervised: Manifold (y1,y2)
to (θ1,θ2)
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Formal and Naïve Representations
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Discovered (Naïve) Representation
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Motion Planning
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Motion planning
Given start / goal image, map to manifold using local interpolation
Use k-nn connectivity in manifold as “roadmap” for motion planning
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Obstacle modeling by node deletion
If obstacle intersects robot in image space → delete corresponding nodes from “visual roadmap”
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Path planning as obstacle moves
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Constrained Motion
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Obstacle modeling by node deletion
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Path planning as obstacle moves
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Obstacle modeling by node deletion
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Mobile robots
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Residual error : disk robot
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Robot Motion Planning
Source
Destination
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Path Planning Interface
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Real robots
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SCARA arm
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SCARA arm : degrees of freedom
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Background Subtraction : Robotimage
learned background
foreground (moving part)
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Background Subtraction : Obstacle
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Visual Configuration Space
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5-DOF Adept CRS manipulator
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Application to Graphics
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Head Motion
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Conclusion
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Beyond Geometry
• Geometry is not everything!!
• Real robots have limitations on acceleration owing to torque / inertia Dynamics
• Learning to plan motions?
- Babies learn to move arms
- Learn low-dimensional representations of motion
• Grasping / Assembly : Motions along obstacle boundary
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Humans and Robots
madhur ambastha cs665 2002