EE474 Robotics and Control

Lecture 3: Kinematics

Dr Gordon Dobie, EEE

Suggested Book
• Introduction to Autonomous
Mobile Robots
• by Roland Siegwart , Illah R.
Nourbakhsh, Davide Scaramuzza
• ISBN-10: 0262015358
• ISBN-13: 978-0262015356

• Material from Chapter 3

Suggested Book 2

• Where am I? Sensors and

Methods for Mobile Robot
Positioning
• By: Borenstein, H.R. Everett, L
Feng
• Available online as a pdf.

• An easy to understand
kinematic model – see
bottom of page 19
Wheeled Robot - Kinematics
• Description of mechanical behavior of the robot for
design and control
• Similar to robot manipulator kinematics (Semester 1)
• However, mobile robots can move unbound with
respect to its environment
– there is no direct way to measure the robot’s position
– Position must be integrated over time
– Leads to inaccuracies of the position (motion) estimate
-> the number 1 challenge in mobile robotics
• Understanding mobile robot motion starts with
understanding wheel constraints placed on the robots
mobility
Coordinate Transforms

• Consult your old Maths textbooks, or:

• http://www.continuummechanics.org/coord
xforms.html
 Representing Robot Position
• Representing to robot within an arbitrary initial frame
– Initial frame: X I , YI 
Y I

– Robot frame: X , Y  Y R
R R

– Robot position:  I  x y  
T XR


P
– Mapping between the two frames
–   R   R   x y T XI
R I
AMR: 3.1
 cos sin  0
R    sin  0
YI
cos XR

 0 0 1
– Example: Robot aligned with YI YR


 Xi (Greek c) XI

AMR: 3.2
Kinematics Model
 T
• establish the robot speed   x y  as a function of the wheel
speeds  i , steering angles  i , steering speeds  and the i
geometric parameters of the robot (configuration coordinates).
• forward kinematics y I
 x  v(t)

   y   f ( 1 , n , 1 ,  m , 1 ,  m ) s(t)


 
• Inverse kinematics
xI

 
1   n 1   m 1   m 
T
 f ( x , y ,) AMR: 3.3
Example

AMR: Fig 3.2

 Wheel Kinematic Constraints:
Assumptions Y I

• Movement on a horizontal plane YR

• Point contact of the wheels
• Wheels not deformable XR
• Pure rolling
• No slipping, skidding or sliding 
• No friction for rotation around P
contact point
• Steering axes orthogonal to the XI
surface
• Wheels connected by rigid   r
frame (chassis)
v
Differential Drive Kinematic Model
Adapted from Where Am I? Book

 = encoder counts in interval

X = position (x,y,z)
D = wheel diameter
R = encoder counts
Wheel Kinematic Constraints:

Fixed Standard Wheel

Standard wheel: Two degrees of freedom;

rotation around the (motorized)
wheel axle and the contact point

AMR: Fig 3.3

Fixed Standard Wheel (Rolling)
Fixed Standard Wheel (Rolling)
Fixed Standard Wheel (Rolling)
Fixed Standard Wheel (Slip)
Wheel Kinematic Constraints:

Fixed Standard Wheel

𝑠𝑖𝑛 𝛼 + 𝛽 −𝑐𝑜𝑠 𝛼 + 𝛽 −𝑙 cos 𝛽 𝑅 𝜃 𝜉 ̇ − 𝑟𝜑̇ = 0

𝑐𝑜𝑠 𝛼 + 𝛽 𝑠𝑖𝑛 𝛼 + 𝛽 𝑙𝑠𝑖𝑛 𝛽 𝑅 𝜃 𝜉 ̇ = 0

AMR: Fig 3.3

Example

• Suppose that the wheel A is in position such that

•  = 0 and  = 0
• This would place the contact point of the wheel on XI with
the plane of the wheel oriented parallel to YI. If  = 0, then
the sliding constraint reduces to:
Odometry Model
[See Differential Drive Proof (pdf) on myplace]
Odometry Model DDR
Odometry Model DDR
Odometry Model DDR
Odometry Model DDR
Odometry Model DDR
Wheel Kinematic Constraints:

Steered Standard Wheel

𝑠𝑖𝑛 𝛼 + 𝛽 𝑡 −𝑐𝑜𝑠 𝛼 + 𝛽 𝑡 −𝑙 cos 𝛽 𝑡 𝑅 𝜃 𝜉 ̇ − 𝑟𝜑̇ = 0

𝑐𝑜𝑠 𝛼 + 𝛽 𝑡 𝑠𝑖𝑛 𝛼 + 𝛽 𝑡 𝑙𝑠𝑖𝑛 𝛽 𝑡 𝑅 𝜃 𝜉̇ = 0

AMR: Fig 3.5

Wheel Kinematic Constraints:

Castor Wheel
𝑠𝑖𝑛 𝛼 + 𝛽 𝑡 −𝑐𝑜𝑠 𝛼 + 𝛽 𝑡 −𝑙 cos 𝛽 𝑡 𝑅 𝜃 𝜉 ̇ − 𝑟𝜑̇ = 0

𝑐𝑜𝑠 𝛼 + 𝛽 𝑡 𝑠𝑖𝑛 𝛼 + 𝛽 𝑡 𝑑 + 𝑙𝑠𝑖𝑛 𝛽 𝑡 𝑅 𝜃 𝜉 ̇ + 𝑑 𝛽̇ = 0

castor wheel: Two degrees of freedom;

rotation around an offset steering
joint

AMR: Fig 3.6

 Swedish wheel: Three degrees of
freedom; rotation around the
Wheel Kinematic Constraints: (motorized)
wheel axle, around the rollers and
Swedish Wheel around the contact point

𝑠𝑖𝑛 𝛼 + 𝛽 + 𝛾 −𝑐𝑜𝑠 𝛼 + 𝛽 + 𝛾 −𝑙 cos 𝛽 + 𝛾 𝑅 𝜃 𝜉 ̇

− 𝑟𝜑̇ 𝑐𝑜𝑠𝛾 = 0
𝑐𝑜𝑠 𝛼 + 𝛽 + 𝛾 𝑠𝑖𝑛 𝛼 + 𝛽 + 𝛾 𝑙𝑠𝑖𝑛 𝛽 + 𝛾 𝑅 𝜃 𝜉 ̇ − 𝑟𝜑̇ 𝑠𝑖𝑛𝛾
−𝑟 𝜑 ̇ =0

AMR: Fig 3.8

Wheel Kinematic Constraints:

Spherical Wheel

AMR: Fig 3.9

Robot Kinematic Constraints
• Given a robot with M wheels
– each wheel imposes zero or more constraints on the robot
motion
– only fixed and steerable standard wheels impose constraints
• What is the maneuverability of a robot considering a
combination of different wheels?
• Suppose we have a total of N=Nf + Ns standard wheels
– We can develop the equations for the constraints in matrix forms:
– Rolling
  f (t )  J1 f 
J1 (  s ) R( ) I  J 2  0  (t )  

 J1 (  s )    J 2  diag (r1  rN )
 s (t )  J (
 1s s   )
– Lateral movement N f  N s 1 N f  N s 3

C1 (  s ) R( )I  0  C1 f 
C1 (  s )   
C ( 
 1s s )
N f  N s 3
Mobile Robot Maneuverability
• The maneuverability of a mobile robot is the combination
– of the mobility available based on the sliding constraints
– plus additional freedom contributed by the steering

• Three wheels is sufficient for static stability

– additional wheels need to be synchronised
– this is also the case for some arrangements with three wheels

• It can be derived using the equation seen before

– Degree of mobility  m
– Degree of steerability  s
– Robots maneuverability  M   m   s
Mobile Robot Maneuverability:
Degree of Mobility
• To avoid any lateral slip the motion vector R( )I has to satisfy
the following constraints:
C1 f R( )I  0  C 
C1 (  s )   1 f 
C1s (  s ) R( )I  0 C1s (  s )

• Mathematically:
– R( )I must belong to the null space of the projection matrix C1 (  s )
– Null space of C1 (  s ) is the space N such that for any vector n in
N
C1 (  s )  n  0

– Geometrically this can be shown by the Instantaneous Center of

Rotation (ICR)
Mobile Robot Maneuverability: Instantaneous Center
of Rotation

• Ackermann Steering Bicycle

AMR: Fig 3.12

Mobile Robot Maneuverability: More on Degree of
Mobility

• Robot chassis kinematics is a function of the set of

independent constraints rank C1 (  s )
– the greater the rank of , C1 (  s ) the more constrained is the
mobility
 m  dim N C1 (  s )  3  rank C1 (  s )
• Mathematically 0  rank C1 (  s )  3

• no standard wheels rank C1 (  s )  0

• all direction constrained rank C1 (  s )  3
• Examples:
– Unicycle: One single fixed standard wheel
– Differential drive: Two fixed standard wheels
• wheels on same axle rank C1 (  s )  1
• wheels on different axle rank C1 (  s )  1

N.B. Rank of a matrix is the largest number of linearly independent rows and columns
Mobile Robot Maneuverability:
Degree of Steerability
• Indirect degree of motion
 s  rank C1s (  s )
– The particular orientation at any instant imposes a kinematic
constraint
– However, the ability to change that orientation can lead
additional degree of maneuverability
• Range of  s : 0   s  2

• Examples:
– one steered wheel: Tricycle rank C1s (  s )  1
– two steered wheels: No fixed standard wheel rank C1s (  s )  2
– car (Ackermann steering): Nf = 2, Ns=2 -> common axle
rank C1s (  s )  1
Mobile Robot Maneuverability:
Robot Maneuverability
• Degree of Maneuverability
M  m s

– Two robots with same  M are not necessary equal

– Example: Differential drive and Tricycle (next slide)

– For any robot with  M  2 the ICR is always

constrained
to lie on a line
– For any robot with  M  3 the ICR is not
constrained an
can be set to any point on the plane
Mobile Robot Maneuverability:
Wheel Configurations
• Differential Drive Tricycle

AMR: Fig 3.13

Five Basic Types of Three-Wheel
Configurations

AMR: Fig 3.14

Mobile Robot Workspace:
Degrees of Freedom
• Maneuverability is equivalent to the vehicle’s degree of
freedom (DOF)
• But what is the degree of vehicle’s freedom in its
environment?
– Car example
• Workspace
– how the vehicle is able to move between different
configuration in its workspace?
• The robot’s independently achievable velocities
– = differentiable degrees of freedom (DDOF) =  m
– Bicycle:  M   m   s  1  1 DDOF = 1; DOF=2
– Omni Drive:  M   m   s  3  0 DDOF=3; DOF=3
Mobile Robot Workspace: Degrees of Freedom,
Holonomy

• DOF degrees of freedom:

– Robots ability to achieve various poses
• DDOF differentiable degrees of freedom:
– Robots ability to achieve various path
DDOF   M  DOF
• Holonomic Robots
– A holonomic kinematic constraint can be expressed a an
explicit function of position variables only
– A non-holonomic constraint requires a different
relationship, such as the derivative of a position variable
– Fixed and steered standard wheels impose non-holonomic
constraints
Path / Trajectory Considerations:
Omnidirectional Drive

AMR: Fig 3.15

Path / Trajectory Considerations: Two-
Steer

AMR: Fig 3.16

What You Should Have Learnt!

• A mathematical framework to calculate:

– A kinematic model for a wheeled robot
– The degrees of mobility
– The degrees of steerability
– The degrees of manoeuvrability

– You can now do the lab part 1 and 2.