Kinematics of Rigid Body Transforms and Kinematic Chains in Robotics, Lecture notes of Kinematics

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Kinematics, Kinematics Chains

Previously

• Representation of rigid body motion
• Two different interpretations
  • as transformations between different

coordinate frames

• as operators acting on a rigid body
• Representation in terms of homogeneous

coordinates

• Composition of rigid body motions
• Inverse of rigid body motion

Kinematic Chains

• We will focus on mobile robots (brief digression)
• In general robotics - study of multiple rigid bodies

lined together (e.g. robot manipulator)

• Kinematics – study of position, orientation,

velocity, acceleration regardless of the forces

• Simple examples of kinematic model of robot

manipulator and mobile robot

• Components – links, connected by joints

Various joints

Forward kinematics for a 2D arm

• Find position of the end effector as a function of

the joint angles

• Blackboard example

Kinematic Chains in 3D

• More joints possible (spherical, screw)
• Additional offset parameters, more complicated
• Same idea: set up frame with each link
• Define relationship between links
• Two rules:
  • use Z-axis as an axis of a revolute joint
  • connect two axes shortest distance

In 2D we need only link length and joint angle to

specify the transform

In 3D Denavit-Hartenberg

parameters (see LaValle (chapter [3])

Jacobians

• Kinematics enables us study what space is reachable
• Given reachable points in space, how well can be motion of

an arm controlled near these points

• We would like to establish relationship between velocities

in joint space and velocities in end-effector space

• Given kinematics equations for two link arm
• The relationship between velocities is
• manipulator Jacobian

Manipulator Jacobian

• Determinant of the Jacobian
• If determinant is 0, there is a singularity
• Manipulator kinematics: position of end effector

can be determined knowing the joint angles

• Actuators: motors that drive the joint angles
• Motors can move the joint angles to achieve

certain position

• Mobile robot actuators: motors which drive the

wheels

• Configuration of a wheel does not reveal the pose

of the robot, history is important

Mobile robot kinematics

• Depends on the type of robot

Position and type of the wheels

Two types of wheels

a) Standard – rotation around

(motorized) wheel axel and

the contact point

b) Castor wheel – rotation around

wheel axes, contact point and

castor axel

c) Swedish wheels

d) Ball wheels

• Representing to robot within an arbitrary initial frame
  • Initial frame:
  • Robot frame:
  • Robot pose:
  • Mapping between the two frames
  • transforms points/velocities from body to inertial

frame

• Example: Robot aligned with Y I

Representing Mobile Robot Position

€

T ( θ , x , y ) =

cos θ −sin θ x

sin θ cos θ y

0 0 1

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

Mobile Robot Kinematic Models

• Manipulator case – given joint angles, we can

always tell where the end effector is

• Mobile robot basis – given wheel positions we

cannot tell where the robot is

• We have to remember the history how it got there
• Need to find relationship between velocities and

changes in pose

• Presented on blackboard (see handout)
• How is the wheel velocity affecting velocity of the

chassis

Differential Drive Kinematics

• Blackboard derivation
• Kinematics in the robot frame
• Relationship between robot frame and inertial

frame

x ˙

y ˙

θ

R

v l +v r

2

vr −vl

l

v

ω

x ˙

y ˙

θ

R

cos θ sin θ 0

− sin θ cos θ 0

x ˙

y ˙

θ

I