Robotics

Manipulators P

R R

R R

Robotics R

Chapter 25 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.

Chapter 25

1

Outline

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4

Non-holonomic robots

Robots, Effectors, and Sensors Localization and Mapping θ

Motion Planning (x, y)

Motor Control A car has more DOF (3) than controls (2), so is non-holonomic; cannot generally transition between two infinitesimally close configurations

Chapter 25

Chapter 25

2

5

Sensors

Mobile Robots

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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3

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6

Localization—Where Am I?

Localization contd.

Compute current location and orientation (pose) given observations: At−2

At−1

Can also use extended Kalman filter for simple cases: robot

At

Xt−1

Xt

Xt+1 landmark

Assumes that landmarks are identifiable—otherwise, posterior is multimodal Zt−1

Zt

Zt+1

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Localization contd.

10

Mapping Localization: given map and observed landmarks, update pose distribution

ω t ∆t

xi, yi

Chapter 25

Mapping: given pose and observed landmarks, update map distribution θt+1

h(xt)

vt ∆t

SLAM: given observed landmarks, update pose and map distribution Z1

Z2

Z3

Z4

Probabilistic formulation of SLAM: add landmark locations L1, . . . , Lk to the state vector, proceed as for localization

xt+1 θt

xt

Assume Gaussian noise in motion prediction, sensor range measurements

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Localization contd.

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Mapping contd.

Can use particle filtering to produce approximate position estimate

Robot position

Robot position Robot position

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9

3D Mapping example

Cell decomposition example goal start goal start

Problem: may be no path in pure freespace cells Solution: recursive decomposition of mixed (free+obstacle) cells

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13

Motion Planning

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17

Skeletonization: Voronoi diagram

Idea: plan in configuration space defined by the robot’s DOFs

Voronoi diagram: locus of points equidistant from obstacles

conf-2 conf-1

conf-3

conf-3 conf-2

conf-1 w elb

w shou

Solution is a point trajectory in free C-space Problem: doesn’t scale well to higher dimensions

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14

Configuration space planning

Skeletonization: Probabilistic Roadmap A probabilistic roadmap is generated by generating random points in C-space and keeping those in freespace; create graph by joining pairs by straight lines

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/lines that form a graph such that any two points are connected by a path on the graph

Problem: need to generate enough points to ensure that every start/goal pair is connected through the graph Chapter 25

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18

Motor control

Simple learning algorithm: Stochastic gradient

Can view the motor control problem as a search problem in the dynamic rather than kinematic state space: – state space defined by x1, x2, . . . , x˙1, x˙2, . . . – continuous, high-dimensional (Sarcos humanoid: 162 dimensions)

Minimize Eθ [y 2] by gradient descent: Z

∇θ0 Eθ [y 2] = ∇θ0 Pθ0 (θ)F (θ)2dθ Z ∇ P (θ) θ0 θ0 F (θ)2Pθ0 (θ)dθ = Pθ0 (θ) ∇θ Pθ (θ) = Eθ [ 0 0 y 2 ] Pθ0 (θ)

Deterministic control: many problems are exactly solvable esp. if linear, low-dimensional, exactly known, observable Simple regulatory control laws are effective for specified motions

Given samples (θj , yj ), j = 1, . . . , N , we have

Stochastic optimal control: very few problems exactly solvable ⇒ approximate/adaptive methods

1 N

∇θ0 Eˆθ [y 2] =

N X

j=1

∇θ0 Pθ0 (θj ) 2 y Pθ0 (θj ) j

For Gaussian noise with covariance Σ, i.e., Pθ0 (θ) = N (θ0, Σ), we obtain 1 N

∇θ0 Eˆθ [y 2] =

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N X

j=1

Σ−1(θj − θ0)yj2

19

Biological motor control

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What the algorithm is doing

Motor control systems are characterized by massive redundancy

x

x

Infinitely many trajectories achieve any given task

x

x x x x x x x xx x x x x xx x x x x x x x

E.g., 3-link arm moving in plane throwing at a target simple 12-parameter controller, one degree of freedom at target 11-dimensional continuous space of optimal controllers

x

x

x x

x x x x xx x x x x x x x

x

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 noise explains eye saccade velocity profile perfectly

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Setup

Results for 2–D controller

Suppose a controller has “intended” control parameters θ0 which are corrupted by noise, giving θ drawn from Pθ0

10 9

Output (e.g., distance from target) y = F (θ); Velocity v

y

8 7 6 5 4 0.2

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0.4

0.6 0.8 Angle phi

1

1.2

Results for 2–D controller 4.61 4.6 4.59 Velocity v

4.58 4.57 4.56 4.55 4.54 4.53 4.52 4.51 0.6

0.61

0.62

0.63

0.64

0.65

Angle phi

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Results for 2–D controller 0.0095 0.009 0.0085 E(y^2)

0.008 0.0075 0.007 0.0065 0.006 0.0055 0

2000

4000

6000

8000

10000

Step

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 space requires some method for finitization