ece5463: introductiontorobotics lecturenote2 ...zhang/roboticsclass/...configurationspa… · 2π...

ECE5463: Introduction to Robotics

Lecture Note 2: Configuration Space

Prof. Wei Zhang

Department of Electrical and Computer EngineeringOhio State UniversityColumbus, Ohio, USA

Spring 2018

Lecture 2 (ECE5463 Sp18) Wei Zhang(OSU) 1 / 19

Outline

• Mechanical Structure of a Robot

• Configuration Space

• Representation of Configuration Space

Outline Lecture 2 (ECE5463 Sp18) Wei Zhang(OSU) 2 / 19

Typical Mechanical Structure

• A robot is mechanically constructed by connecting a set of bodies, calledlinks, to each other using various types of joints.

• Links are usually modeled as rigid bodies

• Actuators, such as electric motors, deliver forces and torques to the joints,thereby causing motion of the robot

• End-effector, such as gripper or hand, is attached to a specific link

Mechanical Structure Lecture 2 (ECE5463 Sp18) Wei Zhang(OSU) 3 / 19

Typical Joints

• Revolute Joint (R):

• Prismatic Joint (P):

• Helical Joint (H):

• Cylindrical Joint (C):

• Universal Joint (U):

• Spherical Joint (S):

Mechanical Structure Lecture 2 (ECE5463 Sp18) Wei Zhang(OSU) 4 / 19

Outline




Configuration Space Lecture 2 (ECE5463 Sp18) Wei Zhang(OSU) 5 / 19

Configuration Space Definitions

• Configuration: a complete specification of the position of every point of therobot.

• Degree of Freedom (dof): The minimum number of real-valuedcoordinates needed to represent the configuration

• Configuration Space (C-space): The space (set) that contains all possibleconfigurations of the robot.

• Effective representation of the C-space is essential for many aspects ofrobotics


How to find the dof?

• Example: coin on a table

A

B

C


DoF of Planar and Spatial Rigid Body

A

B

C

x y

z

•


DoF of Joints

• Joint can be viewed as providing freedoms to allow one rigid body to moverelative to another.

• Dof of a joint: minimum # of variables needed to represent the configurationof a joint

• Joint can also be viewed as providing constraints on the possible motions ofthe two rigid bodies it connects

Constraints c Constraints cbetween two between two

Joint type dof f planar spatialrigid bodies rigid bodies

Revolute (R) 1 2 5Prismatic (P) 1 2 5

Helical (H) 1 N/A 5Cylindrical (C) 2 N/A 4

Universal (U) 2 N/A 4Spherical (S) 3 N/A 3


twotwootwo

DoF of Mechanisms (Linkages)

dof =(sum of freedoms of the bodies)−number of independent constraints (1)

• Grubler’s Formula: dof = m(N − 1− J) +∑J

i=1 fi


d fd fd

DoF Examples


DoF Examples

R

S

R

RS

SS


Outline




Representation Lecture 2 (ECE5463 Sp18) Wei Zhang(OSU) 14 / 19

Issues for Explicit Parameterization

• Representation of Euclidean space: choose reference frame and representpoint as a vector

• Representation of curved space is more tricky than it appears

• Explicit parameterization uses the same number of coordinates as thespace dimension (suffer from singularity)

Sphere S2


Topology

• Explicit parameterization of sphere (latitude/longitude) suffers fromsingularity because sphere and plane have different topologies.

• Roughly, two spaces are topologically equivalent if one can be continuouslydeformed into the other without cutting or gluing.

• Topologically distinct 1-d spaces:

- circle:

- line:

- closed interval:


Topology

• Examples of topologically different 2-d spacessystem topology sample representation

x

y(x, y)

point on a plane E2

R2

longitude

latitude90◦

−90◦−180◦ 180◦

spherical pendulum S2 [−180◦, 180◦)× [−90◦, 90◦]

00

2π

2π θ1

θ2

2R robot arm T 2=S1×S1 [0, 2π)× [0, 2π)

θ

x0

2π......

rotating sliding knob E1 × S1

R1 × [0, 2π)


Implicit Representation of C-Space

• Implicit Representation: View n-dim space as embedded in a higherdimensional Euclidean space subject to constraints.

• Use more coordinates than the minimum, but can avoid singularities

• Example: Sphere S2

• In this class, we will primarily use the implicit representation


Summary Questions

• What is the configuration space (C-space) of a robot?

• What is dof of C-space, and how to find dof?

• What is topological equivalence?

• Pros and cons for explicit and implicit representation of C-space

• Further reading: chapter 2 of Lynch and Park.


ece5463: introductiontorobotics lecturenote2 ...zhang/roboticsclass/...configurationspa… · 2π...

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