Robotics

Chapter 25

Chapter 25 1

Outline

Robots, Effectors, and Sensors

Localization and Mapping

Motion Planning

Motor Control

Chapter 25 2

Mobile Robots

Chapter 25 3

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.

Chapter 25 4

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

Chapter 25 5

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

Chapter 25 6

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

Chapter 25 7

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

Chapter 25 8

Localization contd.

Can use particle filtering to produce approximate position estimate

Robot positionRobot position

Robot position

Chapter 25 9

Localization contd.

Can also use extended Kalman filter for simple cases:

robot

landmark

Assumes that landmarks are identifiable—otherwise, posterior is multimodal

Chapter 25 10

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

Chapter 25 11

Mapping contd.

Chapter 25 12

3D Mapping example

Chapter 25 13

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

Chapter 25 14

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

Chapter 25 15

Cell decomposition example

start

goal

startgoal

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

Chapter 25 16

Skeletonization: Voronoi diagram

Voronoi diagram: locus of points equidistant from obstacles

Problem: doesn’t scale well to higher dimensions

Chapter 25 17

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

Chapter 25 18

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

Chapter 25 19

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

Chapter 25 20

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

Chapter 25 21

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

Chapter 25 22

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

Chapter 25 23

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

Chapter 25 24

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

Chapter 25 25

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

Chapter 25 26

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

Chapter 25 27