1. Mobile Robot Wheel Configurations and Kinematicsx y
w(t)
Mobile Robot Wheel Configurations and Kinematics
V(t) VL 2d
Cmput412
Martin Jagersand
With slides from Zach Dodds, Roland Siegwart, Sebastian Thrun VR
ICC
R(t)

2. Autonomous Mobile Robots
Three key questions in Mobile Robotics
Where am I ?
Where am I going ?
How do I get there ?
To answer these questions the robot has to
have a model of the robot and environment (given or autonomously built)
perceive and analyze the environment
find robot position within the environment
plan and execute the movement
Today: Focus on Where am I going ? using wheel encoders + model

3. Human Navigation: Topological with imprecise metric information
Courtesy K. Arras

4. Human Navigation: Topological with imprecise metric information
Odometry
(Imprecise)
Feature-based Navigation
still a challenge for artificial systems
Corridor
crossing
Elevator door
Entrance
How to find a treasure
Courtesy K. Arras
Eiffel Tower

5. Environment Representation: The Map Categories
Recognizable Locations
Topological Maps
Courtesy K. Arras
Metric Topological Maps
Fully Metric Maps (continuos or discrete)

6. applied-kinematics.com: forensic animation
If we move a wheel one radian, how does the robot center move?
Robot geometry and kinematic analysis gives answers!
from sort of simple to sort of complex
wheeled platforms
manipulator modeling
applied-kinematics.com: forensic animation

7. Kinematics vs. Dynamics
The effect of a robot’s geometry on its motion.
The effect of all forces (internal and external) on a robot’s motion.
If the motors move this much, where will the robot be?
If the motors apply this much force, where will the robot be?
Assumes that we control encoder readings…
Assumes that we control motor current…
consider the segway…

8. Kinematics kinematics dynamics
The effect of a robot’s geometry on its motion.
The effect of all forces (internal and external) on a robot’s motion.
from sort of simple to sort of complex
Aaargh!
two-link manipulator
three-link manipulator
represented by a 4x4 matrix

9. Mobile Robot Different Arrangements of Wheels
Two wheels
Three wheels
Four wheels Six wheels

10. “Quiz” Odometry / Dead Reckoning Where will C be? C A B 40cm
or why optical encoders are so useful…
Suppose a robots’ wheels are arranged as shown below. The robot moves its wheels smoothly and evenly forward (initially +y) until Wheel A has made 8 full rotations and Wheel B has made 6 full rotations. Where is the robot? That is, what are the coordinates of its center, C, if C was initially at (0,0)?
Assume that the wheels do not slip as they move. (A strong assumption...)
Where will C be? y C x A B
40cm
Wheels A and B have a diameter of 5 cm.

11. Differential drive Most common kinematic choice
- difference in wheels’ speeds determines its turning angle
Most miniature robots…
ER1, Pioneer, Roomba VL VR

12. Questions (forward kinematics)
Differential drive
Most common kinematic choice
- difference in wheels’ speeds determines its turning angle
Most miniature robots…
ER1, Pioneer, Roomba
Questions (forward kinematics)
Given the wheel’s velocities or positions, what is the robot’s velocity/position ? VL
Are there any inherent system constraints? VR

13. Questions (forward kinematics)
Differential drive
Most common kinematic choice
- difference in wheels’ speeds determines its turning angle
Most miniature robots…
ER1, Pioneer, Roomba
Questions (forward kinematics)
Given the wheel’s velocities or positions, what is the robot’s velocity/position ? VL
Are there any inherent system constraints? VR
1) Specify system measurements
2) Determine the point (the radius) around which the robot is turning.
3) Determine the speed at which the robot is turning to obtain the robot velocity.
4) Integrate to find position.

14. Differential drive y x VL q 2d VR 1) Specify system measurements
- consider possible coordinate systems VL x q 2d VR

15. Is there always a point around which the robot is rotating?
Differential drive
1) Specify system measurements y
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning.
Is there always a point around which the robot is rotating? VL x q 2d VR

16. Differential drive y x VL q 2d VR same angular velocity??
1) Specify system measurements y
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning.
- to minimize wheel slippage, the instantaneous center of curvature (the ICC) must lie at the intersection of the wheels’ axles VL x q
- each wheel must be traveling at the same angular velocity 2d VR
same angular velocity??

17. Differential drive y x w VL q 2d VR R
1) Specify system measurements y
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. w
- to minimize wheel slippage, this point (the ICC) must lie at the intersection of the wheels’ axles VL x
- each wheel must be traveling at the same angular velocity around the ICC q 2d VR
ICC “instantaneous center of curvature” R
robot’s turning radius
How are these values related?
(the wheel diameter is already accounted for in VL and VR )

18. Differential drive y x w(R+d) = VL w(R-d) = VR w VL 2d VR R
1) Specify system measurements x y
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. w
- each wheel must be traveling at the same angular velocity around the ICC VL
3) Determine the robot’s speed around the ICC and its linear velocity 2d VR
w(R+d) = VL
ICC R
w(R-d) = VR
robot’s turning radius

19. Differential drive y x w(R+d) = VL w(R-d) = VR w VL 2d VR R
1) Specify system measurements x y
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. w
- each wheel must be traveling at the same angular velocity around the ICC VL
3) Determine the robot’s speed around the ICC and its linear velocity 2d VR
w(R+d) = VL
ICC
of these five, what’s known & what’s not? R
w(R-d) = VR
robot’s turning radius

20. are there interesting cases?
Differential drive
1) Specify system measurements x y
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. w
- each wheel must be traveling at the same angular velocity around the ICC VL
3) Determine the robot’s speed around the ICC and its linear velocity 2d VR
w(R+d) = VL
ICC R
w(R-d) = VR
robot’s turning radius
Thus,
w = ( VR - VL ) / 2d
are there interesting cases?
R = d ( VR + VL ) / ( VR - VL )

21. Differential drive y x w(R+d) = VL w(R-d) = VR w = ( VR - VL ) / 2d
1) Specify system measurements x y
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. w
- each wheel must be traveling at the same angular velocity around the ICC V VL
3) Determine the robot’s speed around the ICC and its linear velocity 2d VR
w(R+d) = VL
ICC R
w(R-d) = VR
robot’s turning radius
Thus,
w = ( VR - VL ) / 2d
R = d ( VR + VL ) / ( VR - VL )
So, what is the robot’s velocity?

22. Differential drive y x w(R+d) = VL w(R-d) = VR w = ( VR - VL ) / 2d
1) Specify system measurements x y
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. w
- each wheel must be traveling at the same angular velocity around the ICC V VL
3) Determine the robot’s speed around the ICC and its linear velocity 2d VR
w(R+d) = VL
ICC R
w(R-d) = VR
robot’s turning radius
Thus,
w = ( VR - VL ) / 2d
R = d ( VR + VL ) / ( VR - VL )
So, where’s the robot?
So, the robot’s velocity is
V = wR = ( VR + VL ) / 2

23. Differential drive y x Kinematics Vx = V(t) cos(q(t)) w(t)
4) Integrate to obtain position x y
Vx = V(t) cos(q(t))
w(t)
Vy = V(t) sin(q(t))
Thus,
V(t)
x(t) = ∫ V(t) cos(q(t)) dt VL
y(t) = ∫ V(t) sin(q(t)) dt 2d
q(t) = ∫ w(t) dt VR
ICC
Kinematics
R(t)
robot’s turning radius
with
w = ( VR - VL ) / 2d
R = d ( VR + VL ) / ( VR - VL )
What has to happen to change the ICC ?
V = wR = ( VR + VL ) / 2
things have to change over time, t

24. Q: Two-steered-wheel bicycle
Unusual kinematics:
powered front wheel (velocity = vf )
both wheels (in orientation) can be steered independently: f r
there is a rigid frame between the wheels with length L f
Hw #2: What are the velocity kinematics of this vehicle? r vf L

25. A Lego Mindstorms example! Mecos tricycle-drive robot
Q: Tricycle drive
front wheel is powered and steerable
back wheels tag along (without slipping...)
A Lego Mindstorms example!
Mecos tricycle-drive robot

26. Q: ? The velocity of the front wheel is VF
(i.e., the conversion from angular velocity with wheel diameter is already done…)
We know the distances 2d and B
Mecos tricycle-drive robot VF B VL 2d VR
What else do we need to know?
Where is the ICC?
How fast is the trikebot rotating around it?
What are VL and VR ?

27. Questions (forward kinematics)
Synchro drive
wheels rotate in tandem and remain parallel
Nomad 200
all of the wheels are driven at the same speed
Questions (forward kinematics)
Given the wheel’s velocities or positions, what is the robot’s velocity/position ?
Are there any inherent system constraints?
Where is the ICC ?
1) Specify system measurements
2) Determine the point (the radius) around which the robot is turning.
3) Determine the speed at which the robot is turning to obtain the robot velocity.
4) Integrate to find position.

28. Synchro drive Nomad 200 y Vrobot = Vwheels wrobot = wwheels Kinematics
wheels rotate in tandem and remain parallel
Nomad 200
all of the wheels are driven at the same speed y
ICC at 
Vrobot = Vwheels
velocity w
wrobot = wwheels q
Kinematics x
q(t) = ∫ w(t) dt
Vwheels
x(t) = ∫ Vwheels(t) cos(q(t)) dt
position
y(t) = ∫ Vwheels(t) sin(q(t)) dt
simpler to control, but ...

29. Anything that can be done, can be done with Lego
Lego Synchro
Anything that can be done, can be done with Lego
the lego motto…

30. Four-wheel Steering The kinematic challenges of parallel parking:
wheels have limited turning angles
no in-place rotation
lots of SUVs around
VFL
VFR
VBL
VBR
What has to happen in order for a car's wheels not to slip while turning, i.e., for there to be a single ICC?

31. Each front wheel has to turn a different amount !
Ackerman Steering aL
Similar to a tricycle-drive robot aR y r g =
+ d
VFL
tan(aR) wg =
VFR
VFR
sin(aR)
determines w g
Each front wheel has to turn a different amount ! d
VBL d
VBR x
above – UCI website... r
ICC
6 bar the Mechanisms lab at UCI 31

32. Each front wheel has to turn a different amount !
Ackerman Steering aL
Similar to a tricycle-drive robot aR y r g =
+ d
VFL
tan(aR) wg =
VFR
VFR
sin(aR)
determines w g
Each front wheel has to turn a different amount ! d
VBL d
VBR x r
ICC
but not in the Barbie Jeep…

33. The other wheel velocities are now fixed!
Ackerman Steering aL
Similar to a tricycle-drive robot aR g y r =
+ d
VFL
tan(aR) wg =
VFR
VFR
sin(aR)
determines w g
The other wheel velocities are now fixed! d
VBL wg d =
VFL
sin(aL)
VBR
aL = tan-1(g / (r + d)) x
w(r - d) = VBR
w(r + d) = VBL r
ICC
This is just the cab...

34. The Big Rigs Applications: 5 link trailer 2 controlled angles
Parking two trailers
scary stuff...

35. “pavement-loading” application (no parking, I suppose)
Multiple trailers ?
Double trailers?
"Double-trailer vehicles, including both western doubles and LCVs, had a threefold increased risk of crash involvement compared to single-trailer vehicles, even after adjusting for other variables significantly affecting crash risk, including empty or loaded travel, driver age, hours driving, type of carrier, and whether the carrier operated intrastate or interstate."
automatic control / robotic applications:
“pavement-loading” application (no parking, I suppose)
see
see
animations.aspx
safety factor requirements? 35

36. Parking/reversing with trailers
The Big Rigs
Applications:
5 link trailer
2 controlled angles
Parking/reversing with trailers

37. Nonholonomicity
All of the robots mentioned thus far share an important (if frustrating) property: they are nonholonomic .
- makes it more difficult to navigate between two arbitrary points
- need to resort to techniques like parallel parking

38. Nonholonomicity
All of the robots mentioned thus far share an important (if frustrating) property: they are nonholonomic .
- makes it more difficult to navigate between two arbitrary points
- need to resort to techniques like parallel parking
By definition, a robot is nonholonomic if it can not move to change its pose instantaneously in all available directions.

39. Nonholonomicity
All of the robots mentioned thus far share an important (if frustrating) property: they are nonholonomic .
- makes it more difficult to navigate between two arbitrary points
- need to resort to techniques like parallel parking
By definition, a robot is nonholonomic if it can not move to change its pose instantaneously in all available directions.
differential-drive robots are nonholonomic VL
multiple-trailer rigs are “very” nonholonomic VR

40. Nonholonomicity
All of the robots mentioned thus far share an important (if frustrating) property: they are nonholonomic .
- makes it more difficult to navigate between two arbitrary points
- need to resort to techniques like parallel parking
By definition, a robot is holonomic if it can move to change its pose instantaneously in all available directions.

41. i.e., the robot’s differential motion is unconstrained.
Nonholonomicity
All of the robots mentioned thus far share an important (if frustrating) property: they are nonholonomic .
- makes it more difficult to navigate between two arbitrary points
- need to resort to techniques like parallel parking
By definition, a robot is holonomic if it can move to change its pose instantaneously in all available directions.
i.e., the robot’s differential motion is unconstrained.
Synchro Drive
is the Nomad holonomic?

42. But the Nomad’s differential motion is constrained.
Nonholonomicity
All of the robots mentioned thus far share an important (if frustrating) property: they are nonholonomic .
- makes it more difficult to navigate between two arbitrary points
- need to resort to techniques like parallel parking
By definition, a robot is holonomic if it can move to change its pose instantaneously in all available directions.
But the Nomad’s differential motion is constrained.
Synchro Drive
two DOF are freely controllable; the third is only indirectly accessible

43. The Four Basic Wheels Types
a) Standard wheel: Two degrees of freedom; rotation around the (motorized) wheel axle and the contact point
b) Castor wheel: Three degrees of freedom; rotation around the wheel axle, the contact point and the castor axle

44. The Four Basic Wheels Typesd)
c) Swedish wheel: Three degrees of freedom; rotation around the (motorized) wheel axle, around the rollers and around the contact point
d) Ball or spherical wheel: Suspension technically not solved

45. Uranus, CMU: Omnidirectional Drive with 4 Wheels
Movement in the plane has 3 DOF
thus only three wheels can be independently controlled
It might be better to arrange three swedish wheels in a triangle

46. Holonomic Robots
Navigation is simplified considerably if a robot can move instantaneously in any direction, i.e., is holonomic.
Omniwheels
Mecanum wheels
tradeoffs in locomotion/wheel design
show examples…
if it can be done at all ...

47. in action and “frisbeeing”
Holonomic Designs
in action and “frisbeeing”
Killough Platform
lego logo…

48. Holonomic hype
“The PeopleBot is a highly holonomic platform, able to navigate in the tightest of spaces…”

49. Robot of the Day Rotundus, Inc. of Sweden – spherical robot How?
all internal (proprioceptive) sensing

50. Robot of the Day Rotundus, Inc. of Sweden – spherical robot
Internal, motor-driven pendulum for shifting the center of mass…

51. Probabilistic Kinematics
We may know where our robot is supposed to be, but in reality it might be somewhere else…
Key question:
supposed final pose y
lots of possibilities for the actual final pose
VL (t) x
VR(t)
starting position
What should we do?

52. Running around in squares
MODEL the error in order to reason about it!
Running around in squares 3
Create a program that will run your robot in a square (~2m to a side), pausing after each side before turning and proceeding.
For 10 runs, collect both the odometric estimates of where the robot thinks it is and where the robot actually is after each side. 2 4
You should end up with two sets of 30 angle measurements and 40 length measurements: one set from odometry and one from “ground-truth.” 1
Find the mean and the standard deviation of the differences between odometry and ground truth for the angles and for the lengths – this is the robot’s motion uncertainty model.
start and “end”
This provides a probabilistic kinematic model.

53. Reasons for Motion Errors
different wheel diameters
ideal case
bump
carpet
and many more …

54. Uncertain motion: Probabilistic kinematics
Repeated application of the sensor model for short movements.
For a motion command u, and starting position x’, what is the probability to end at x?
p(x|u,x’) x’ x’ u u

55. Odometry Model
Robot moves from to
Odometry information

56. The atan2 Function
Extends the inverse tangent and correctly copes with the signs of x and y.

57. Noise Model for Odometry
The measured motion is given by the true motion corrupted with noise.

58. Typical Distributions for Probabilistic Motion Models
Normal distribution
Triangular distribution

59. Calculating the Probability (zero-centered)
For a normal distribution
For a triangular distribution
Algorithm prob_normal_distribution(a,b):
return
Algorithm prob_triangular_distribution(a,b):
return

60. Calculating the Posterior Given x, x’, and u
Algorithm motion_model_odometry(x,x’,u)
return p1 · p2 · p3
odometry values (u)
values of interest (x,x’)

61. Application p(x|u,x’) x’ x’ u u
Repeated application of the sensor model for short movements.
Typical banana-shaped distributions obtained for 2d-projection of 3d posterior.
p(x|u,x’) x’ x’ u u

62. Sample-based Density Representation

63. Sample-based Density Representation

64. How to Sample from Normal or Triangular Distributions?
Sampling from a normal distribution
Sampling from a triangular distribution
Algorithm sample_normal_distribution(b):
return
Algorithm sample_triangular_distribution(b):
return

65. Normally Distributed Samples

66. For Triangular Distribution
103 samples
104 samples
105 samples
106 samples

67. Rejection Sampling Sampling from arbitrary distributions
Algorithm sample_distribution(f,b):
repeat
until ( )
return

68. Example
Sampling from

69. Sample Odometry Motion Model
Algorithm sample_motion_model(u, x):
Return
sample_normal_distribution

70. Sampling from Our Motion Model
Start

71. Examples (Odometry-Based)

72. Holonomic reality Discover Magazine -- Top 10 Innovation
Discover top 10 in 1997
Sage -- a museum tour guide
easier to define than to determine… 72

73. Mobile Robot Examples Automatic Guided Vehicles
Newest generation of Automatic Guided Vehicle of VOLVO used to transport motor blocks from on assembly station to an other. It is guided by an electrical wire installed in the floor but it is also able to leave the wire to avoid obstacles. There are over 4000 AGV only at VOLVO’s plants.

74. Helpmate
HELPMATE is a mobile robot used in hospitals for transportation tasks. It has various on board sensors for autonomous navigation in the corridors. The main sensor for localization is a camera looking to the ceiling. It can detect the lamps on the ceiling as reference (landmark).

75. BR700 Cleaning Robot
BR 700 cleaning robot developed and sold by Kärcher Inc., Germany. Its navigation system is based on a very sophisticated sonar system and a gyro.

76. The Pioneer
Picture of Pioneer, the teleoperated robot that is supposed to explore the Sarcophagus at Chernobyl

77. The Pioneer
PIONEER 1 is a modular mobile robot offering various options like a gripper or an on board camera. It is equipped with a sophisticated navigation library developed at Stanford Research Institute (SRI).

78. The Khepera Robot
KHEPERA is a small mobile robot for research and education. It sizes only about 60 mm in diameter. Additional modules with cameras, grippers and much more are available. More then 700 units have already been sold (end of 1998). K-Team.html

79. Sojourner, First Robot on Mars
The mobile robot Sojourner was used during the Pathfinder mission to explore the mars in summer It was nearly fully teleoperated from earth. However, some on board sensors allowed for obstacle detection nasa.gov/telerobotic s_page/telerobotics. shtm

80. SHRIMP, a Mobile Robot with Excellent Climbing Abilities
Objective
Passive locomotion concept for rough terrain
Results: The Shrimp
6 wheels
one fixed wheel in the rear
two boogies on each side
one front wheel with spring suspension
robot sizing around 60 cm in length and 20 cm in height
highly stable in rough terrain
overcomes obstacles up to 2 times its wheel diameter

81. The SHRIMP Adapts Optimally to Rough Terrain

82. Locomotion Concepts: Principles Found in Nature
Pour commencer, penchons nous sur les principes que l'on trouve dans la nature. On peut caractériser les différents types de locomotion des animaux selon le millieu dans lequel ils évoluent. Dans l'ordre chronologique, la nature a tout d'abord développé des créatures capables de se déplacer dans l'eau ou ce sont les forces hydrodynamiques qui induisent le mouvement à l'animal. Soit par déformation soit par propultion. Les créatures terrestres sont ensuite apparues. Les principes de locomotion terrestres sont basés sur les forces résultant de la gravitation qui sont la force de réaction normale au sol ainsi que les forces de frictions. On peut distinguer plusieurs mécanismes. Tout d'abord il y'a les animaux rampant qui se déplacent en créant des ondes de déformation le long de leur corps qui peuvent être longitudinales (chenilles) ou transversales (serpants). Il y'a ensuite les créatures à pattes qui peuvent se déplacent en faisant osciller leurs membres que l'on peut alors comparer à

83. Walking or rolling? number of actuators structural complexity
control expense
energy efficient
terrain (flat ground, soft ground, climbing..)
movement of the involved masses
walking / running includes up and down movement of COG
some extra losses
Faut-il donc rouler ou marcher ?
En régle générale, les systèmes qui marchent nécessitent beaucoup plus de degrés de libertés et d'actuateurs que les systèmes qui roulent et sont donc beaucoup plus compliqués à mettre en oeuvre.
Les systèmes roulants, en plus de leur simplicité de mise en oeuvre, permettent d'atteindre de grandes vitesses avec de très bons rendements pour autant que le terrain soit plat. On voit sur ce graphique que les systèmes roulants sont de une à deux fois meilleurs que les systèmes marchants. Les pertes dans un système roulant sont principalement dues à la resistance au roulement. Cette dernière est caractérisée par le type de pneu et le type de terrain sur lequel la roue se déplace. On constate que le cas idéal est celui du train ou une roue en acier roule sur un rail en acier. Cette technologie permet de déplacer de très grosses charges a des vitesses très élevées. La roue trouve ses limites dans les terrains mous ou parsemés d'obstacles. Dans ces cas la, on constate que les systèmes marchants sont tout aussi efficaces.
Dans les systèmes marchants les pertes sont principalement dues au fait que le centre de gravité monte et descent de facon périodique. Une autre cause sont les chocs répétitifs avec le sol.
Du a la structure de l'environnement qui nous entoure on peut maintenant mieux comprendre que la nature ait choisit la marche comme moyen de locomotion le plus répendu.

84. Forester Robot
Pulstech developed the first ‘industrial like’ walking robot. It is designed moving wood out of the forest. The leg coordination is automated, but navigation is still done by the human operator on the robot.

85. The Honda Walking Robot http://www.honda.co.jp/tech/other/robot.html
Image of Honda Robot

86. ROV Tiburon Underwater Robot
Picture of robot ROV Tiburon for underwater archaeology (teleoperated)- used by MBARI for deep-sea research, this UAV provides autonomous hovering capabilities for the human operator.

87. i.e., designing winged robots Harvard’s Flying Microrobots
Kinematics lives!
i.e., designing winged robots
Harvard’s Flying Microrobots
Univ. of Delaware

88. Kinematics lives! i.e., getting a robotic octopus to move along a path
Modeling interactions
A single tentacle…
The result

89. This will have to wait until well after reading week…
Robot Manipulators
Is this robot holonomic ?
This will have to wait until well after reading week…