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ME5751�Robotics Motion Planning

Yizhe Chang chang@cpp.edu

Lecture Note Set #6-8

Partial lecture slide from

C. Clark�Harvey Mudd College, Claremont, CA�D. J. Cappelleri, M. Salman

Stevens I.T., Hoboken, NJ

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Outline

Locomotion and kinematics
Wheel Kinematics (Crazy Maths!)
Robotics Wheel Constraints
Robot Maneuverability

"Wisdom and virtue are like the two wheels of a cart."

– Japanese Proverb

"Put your shoulder to the wheel."�– Aesop

"Sometimes the wheel turns slowly, but it turns.“ – Lorne Michaels

"Don't reinvent the wheel, just realign it."– Anthony D'Angelo

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Locomotion

Locomotion is the act of moving from place to place.�
Locomotion relies on the physical interaction between the vehicle and its environment. �
Locomotion is concerned with the interaction forces, along with the mechanisms and actuators that generate them

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Locomotion - Issues

Stability

Number of contact points
Center of gravity
Static vs. Dynamic stabilization
Inclination of terrain

Contact

Contact point or area
Angle of contact

Environment

Structure
Medium

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Locomotion in Nature

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Locomotion in Robots

Many locomotion concepts are inspired by nature
Most natural locomotion concepts are difficult to imitate technically
Rolling, which is NOT found in nature, is most efficient

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Sliding locomotion

Snake-like Robot

https://www.youtube.com/watch?v=Kf-budiIf9M

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Crawling locomotion

Crawling locomotion

https://youtu.be/x579QKA6fkY?t=53

Columbia Creative Lab

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Quadrotors locomotion

https://www.youtube.com/watch?time_continue=30&v=_sUeGC-8dyk

Upenn GRASP Lab

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Legged Locomotion

Nature inspired
The movement of walking biped is close to rolling

Number of legs determines stability of locomotion

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Legged Locomotion

Human Leg: degree of freedom?

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Legged Locomotion

Human Leg: degree of freedom?

L = 3�Hip Joint Remove 3 dof

Knee Remove 5 dof

Total dof = 6(L-1)-3-5

= 4

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Legged locomotion

Degrees of freedom per leg

The more dof, the more flexible (complex) motion the leg can do
The more dof, the less stable

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Legged locomotion

Walking gaits
The gait is the repetitive sequence� of leg movements to allow� locomotion
The gait is characterized by the� sequence of lift and release� events of individual legs.

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Legged Locomotion

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Legged locomotion

https://www.youtube.com/watch?v=LikxFZZO2sk

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Wheeled locomotion

Wheel types
A) Standard wheel

2 dof

B) Castor wheel

3 dof

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Wheeled locomotion

Wheel types
C) Swedish wheel

3 dof

D) Ball wheel

3 dof

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Wheeled locomotion

Wheel Arrangements

Three issues: Stability, Maneuverability and Controllability
Stability is guaranteed with 3 wheels, improved with four.
Tradeoff between Maneuverability and Controllability

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Outline

Locomotion and kinematics
Wheel Kinematics (Crazy Maths!)

Fixed standard wheel
Steer standard wheel
Swedish wheel
Spherical wheel

Robotics Wheel Constraints
Robot Maneuverability

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Differential driving

X_I

Y_I

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Wheel Kinematics

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Fixed standard wheel

We have a fixed wheel:
Figure in x_Ry_R – robot frame!

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Fixed standard wheel

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Fixed standard wheel

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Fixed standard wheel SNS p64 (eq 3.12)

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Fixed standard wheel

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Fixed standard wheel

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Fixed standard wheel

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Steered standard wheel

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Steered standard wheel

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Castor wheel

Same as steered, except a fixed d is introduced

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Castor wheel

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Swedish wheel

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Swedish wheel

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Spherical wheel

A ball wheel, how many more dof?

v_A has no fixed direction!

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Outline

Locomotion and kinematics
Wheel Kinematics (Crazy Maths!)
Robotics Wheel Constraints (SNS3.2.5)
Robot Maneuverability

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Robot kinematic constraint

If we have N wheels, with fixed and steerable wheels, can we unify all previous equations into one? (e.g. Swedish and fixed standard)

Swedish constraint:

Standard constraint:

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Robot kinematic constraint

The slide is simply a copy of textbook. It is a show off of math representation. Don’t bother if you cannot understand them!

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Robot kinematic constraint

The slide is simply a copy of textbook. It is a show off of math representation. Don’t bother if you cannot understand them!

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Robot kinematic constraint

The slide is simply a copy of textbook. It is a show off of math representation. Don’t bother if you cannot understand them!

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Let’s do an example:

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Example cont.

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Example Cont

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Example Cont

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Example Cont

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Example cont

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How about this?

Differential driving!! HW

X_I

Y_I

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Outline

Locomotion and kinematics
Wheel Kinematics (Crazy Maths!)
Robotics Wheel Constraints
Robot Maneuverability

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Fixed and steerable standard wheels

At the directional perpendicular to velocity, the speed must be 0.

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Castor wheel

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Robot kinematic constraint

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Robot kinematic constraint

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Robot kinematic constraint

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How about this?

X_I

Y_I

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How about this?

X_I

Y_I

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How about this?

X_I

Y_I

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Rank of a matrix

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Number of constraint

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Number of constraint:

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Degree of mobility

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Number of constraint:

Modified, 3 fixed wheels

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Degree of mobility?

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Degree of steerability

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Degree of steerability

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Degree of steerability

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Degree of steerability

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Degree of maneuverability

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DDOF

DDOF: differentiable degree of freedom
It is always equal to mobility
CMU 3 omnidirectional robot (Swedish wheels)

Mobility: 3
Steerability 0
DDOF: 3

https://www.youtube.com/watch?time_continue=48&v=t1GVBouubww

https://www.societyofrobots.com/robot_omni_wheel.shtml

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Holonomic

Holonomic refers to the relationship between controllable and total degrees of freedom of a robot. If the controllable degree of freedom is equal to total degrees of freedom, then the robot is said to be Holonomic
If the controllable degree of freedom is less than the total degrees of freedom, then it is known as non-Holonomic drive.
A car has three degrees of freedom; i.e. its position in two axes and its orientation. However, there are only two controllable degrees of freedom which are acceleration (or braking) and turning angle of steering wheel.

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