locomotion & kinematics

LOCOMOTION & KINEMATICS

Today

•We’ll start on the bottom right hand side.

cisc3415-fall2011-parsons-lect02 2

Motion

• Two aspects:

– Locomotion– Kinematics

• Locomotion: What kinds of motion are possible?

• Locomotion: What physical structures are there?

• Kinematics: Mathematical model of motion.

• Kinematics: Models make it possible to predict motion.


Locomotion in nature


Sliding

• Snakes have four gaits.

• Lateral undulation(most common)

• Concertina

• Sidewinding

• Rectilinear

• (S5, S3-slither-to-sidewind)


Crawling

• Concertina and Rectilinear motion can be considered crawling.

• Not directly implemented.

• The Makro (left) and Omnitread (right) robots crawl, but notexactly like real snakes do.


Characterisation of locomotion

• Locomotion:

– Physical interaction between the vehicle and its environment.

• Locomotion is concerned with interaction forces, and themechanisms and actuators that generate them.

• The most important issues in locomotion are:

– Stability– Characteristics of contact– Nature of environment


Stability

• Number of contact points

• Center of gravity

• Static/dynamic stabilization

• Inclination of terrain


Characteristics of contact

• Contact point or contact area

• Angle of contact

• Friction


Nature of environment

• Structure

•Medium (water, air, soft or hard ground)


Legged motion• The fewer legs the more complicated locomotion becomes

– at least three legs are required for static stability

• During walking some legs are lifted

• For static walking at least 6 legs are required

– Babies have to learn for quite a while until they are able tostand or walk on their two legs.

mammal (2 or 4) reptile insect


Leg joints

• A minimum of two DOF is required to move a leg forward

– A lift and a swing motion.– Sliding free motion in more than one direction not possible

• Three DOF for each leg in most cases

• Fourth DOF for the ankle joint

– Might improve walking– However, additional joint (DOF) increase the complexity of

the design and especially of the locomotion control.


Leg structure

• How many degrees of freedom does this leg have?


Numbers of gaits

• Gait is characterized as the sequence of lift and release events ofthe individual legs.

– Depends on the number of legs.

• The number of possible events N for a walking machine with klegs is:

N = (2k − 1)!

• For a biped walker (k=2) the number of possible events N is:

N = (4− 1)! = 3! = 3× 2× 1 = 6

• The 6 different events are:lift right leg / lift left leg / release right leg / release left leg / liftboth legs together / release both legs together


• For a robot with 6 legs (hexapod), such as:

• N is alreadyN = 11! = 39, 916, 800


Obvious 6-legged gait

• Static stability— the robot is always stable.

(six-legged-crawl)


Obvious 4-legged gaits

Changeover walk Gallop


Titan VIII

• A family of 9 robots, developed from 1976, to explore gaits.

(Titan walk)


Aibo

(Aibo)


Big Dog, Little Dog

•Work grew out of the MIT Leg Lab.

(leg-lab-spring-flamingo)


Humanoid Robots

• Two-legged gaits are difficult to achieve — human gait, forexample is very unstable.


Nao

• Aldebaran.

(NAO-playtime)


Walking or rolling?

• Number of actuators

• Structural complexity

• Control expense

• Energy efficient

• Terrain (flat ground,soft ground, climbing..)


RHex• Somewhere in between walking and rolling.

(rhex)


Wheeled robots

•Wheels are the most appropriate solution for many applications

– Avoid the complexity of controlling legs

• Basic wheel layouts limited to easy terrain

– Motivation for work on legged robots– Much work on adapting wheeled robots to hard terrain.

• Three wheels are sufficient to guarantee stability

– With more than three wheels a flexible suspension is required

• Selection of wheels depends on the application


Four basic wheels

• Standard wheel: Two degreesof freedom; rotation around the(motorized) wheel axle and thecontact point

• Castor wheel: Three degrees offreedom; rotation around thewheel axle, the contact point andthe castor axle


Four standard wheels (II)

• Swedish wheel: Three degreesof freedom; rotation around the(motorized) wheel axle, aroundthe rollers and around thecontact point.

• Ball (also called spherical) wheel:Omnidirectional. Suspension istechnically not solved.


Swedish wheel

• Invented in 1973 by Bengt Ilon, who was working for theSwedish company Mecanum AB.

• Also called a Mecanum or Ilon wheel.


Characteristics of wheeled vehicles• Stability of a vehicle is be guaranteed with 3 wheels.

– Center of gravity is within the triangle with is formed by theground contact point of the wheels.

• Stability is improved by 4 and more wheel.

– However, such arrangements are hyperstatic and require aflexible suspension system.

• Bigger wheels allow robot to overcome higher obstacles.

– But they require higher torque or reductions in the gear box.

•Most wheel arrangements are non-holonomic (see later)

– Require high control effort

• Combining actuation and steering on one wheel makes thedesign complex and adds additional errors for odometry.


Two wheels

• Steering wheel at front, drive wheel atback.

• Differential drive. Center of massabove or below axle.


• The two wheel differential drive pattern was used in Cye.

• Cye was an early attempt at a robot for home use.


• The Segway RMP has its center of mass above the wheels


Three wheels

• Differential drive plus caster oromnidirectional wheel.

• Connected drive wheels at rear, steeredwheel at front.

• Two free wheels in rear, steered drivewheel in front.


Differential drive plus caster

• Highly maneuverable, but limited to movingforwards/backwards and rotating.


Neptune

• Neptune: an early experimental robot from CMU.


Other three wheel drives

• Omnidirectional

• Three drive wheels.

• Swedish or spherical.

• Synchro drive

• Three drive wheels.

• Can’t control orientation.


Omnidirectional

• The Palm Pilot Robot Kit (left) and the Tribolo (right) (Tribolo)


Synchro• All wheels are actuated synchronously by one motor

– Defines the speed of the vehicle

• All wheels steered synchronously by a second motor

– Sets the heading of the vehicle

(Borenstein)

• The orientation in space of the robot frame will always remainthe same


Four wheels

• Various combinations of steered, driven wheels andomnidirectional wheels.


Four steering wheels

• Highly maneuverable, hard to control.


Nomad

• The Nomad series of robots from Nomadic Teachnologies hadfour casters, driven and steered.


Uranus

• Four omnidirectional driven wheels.

• Not a minimal arrangement, since only three degrees of freedom.

• Four wheels for stability.


Tracked robots

• Large contact area means good traction.

• Use slip/skid steering.


Slip/skid steering

• Also used on ATV versions of differential drive platforms.


Aerial robots

• Lots of interest in the last few years

• Aim is to use them for search and surveillance tasks.

• Starmac (left) and YARB (right)


Fish robots

• Incorporating verison kinds of motion.


Kinematics

• So far we have looked at different kinds of motion in aqualitative way.

• One way to program robots to move is trial and error.

• A somewhat better way is to establish mathematcially how therobot should move, this is kinematics.

• Rather kinematics is the business of figuring how a robot willmove if its motors work in a given way.

• Inverse-kinematics then tells us how to move the motors to getthe robot to do what we want.

•We’ll look at two tiny bits of the kinematics world.


A formal model

•We will assume, as people usualy do, that the robot’s location isfixed in terms of three coordinates:

(xI, yI, θ)

• Given that the robot needs to navigate to a location (xG, yG, θG), itcan determine how x, y and θ need to change.

– BUT it can’t control these directly.


• All it has access to are the speeds of its wheels:

ϕ1, . . . ϕn

The steering angle of the steerable wheels:

β1, . . . βm

and the speed with which those steering angles are changing.

β1, . . . βm

• Together these determine the motion of the robot:

f (ϕ1, . . . ϕn, β1, . . . βm, β1, . . . βm) =

xI

yI

θ

• Forward kinematics


• This is not what we want.

• But we can get what we want from it:

ϕ1...ϕn

β1...βm

β1...βm

= f (xI, yI, θ)

• Reverse kinematics


Representing robot position

• The robot knows how it moves relative to its center of rotation.

• This is not the same as knowing how it moves relative to theworld.

Two systems of coordinates:

– Initial Frame: {XI,YI}– Robot Frame: {XR,YR}


• Robot position:

ξI = [xI, yI, θI]T

•Mapping between frames:

ξR = R(θ)ξI

= R(θ)[xI, yI, θI

]T

where

R(θ) =

cos(θ) sin(θ) 0− sin(θ) cos(θ) 0

0 0 1


• In other wordsxR

yR

θR

= R(θ) =

cos(θ) sin(θ) 0− sin(θ) cos(θ) 0

0 0 1

xI

yI

θI

meaning that:

xR = xI cos(θ) + yI sin(θ) + θI.0

yR = −xI sin(θ) + yI cos(θ) + θI.0

θR = xI.0 + yI.0 + θI.1


• This is the forward kinmatic model, but this still isn’t what wewant — we want the reverse kinematic model:

xI

yI

θI

= R(θ)−1

xR

yR

θR

where R(θ)−1 is the inverse of R(θ).

• Often R(θ)−1 is hard to compute, but luckily for us in this case itisn’t.

•We have:

R(θ)−1 =

cos(θ) − sin(θ) 0sin(θ) cos(θ) 00 0 1

which we can use to establish xI, yI, θI


• In other wordsxI

yI

θI

= R(θ)−1 =

cos(θ) − sin(θ) 0sin(θ) cos(θ) 00 0 1

xR

yR

θR

meaning that:

xI = xR cos(θ)− yR sin(θ) + θR.0

yI = xR sin(θ) + yR cos(θ) + θR.0

θI = xR.0 + yR.0 + θR.1


Down to the structure of the robot

•We can now identify the motion of the robot, in the global frame,if we know:

xR, yR, θ

but how do we tell what these are?


•We compute it from what we can measure like the speed of thewheels:

• Some assumptions — constraints on the motion of the robot:

– Movement on a horizontal plane– Point contact of the wheels; wheels not deformable– Pure rolling, so v = 0 at contact point; no slipping, skidding or

sliding– No friction for rotation around contact point– Steering axes orthogonal to the surface– Wheels connected by rigid frame (chassis)


• Consider differential drive.

•Wheels rotate at ϕ

• Each wheel contributes:rϕ2

to motion of center of rotation.

• Total speed is the sum of twocontributions.

• Rotation due to right wheel is

ωr =rϕ2l

counterclockwise about the leftwheel.l is the distance between wheels.


• Rotation due to the left wheel is:

ωl =−rϕ2l

counterclockwise about the rightwheel.

• Combining these componentswe have:

xR

yR

θR

=

rϕr2 + rϕl

20

rϕr2l −

rϕl2l

• And we can combine these withR(θ)−1 to find motion in theglobal frame.


Wheel geometry

•Making sure the assumptions hold imposes constraints.

• For example, ensuring that the wheel doesn’t slip sayssomething about motion in the direction of the wheel.


More complex scenarios

Steered standard wheel Caster

•More parameters.

• Only fixed and steerable standard wheels impose constraints.


Robot mobility• The sliding constraint means that a standard wheel has no lateral

motion.

– Zero motion line through the axis.

Instantaneouscenter of rotation

• Has to move along a circle whose center is on the zero motionline.


• A differential drive robot has just one line of zero motion.

• Thus its rotation is not constrained

– It can move in any circle it wants.

•Makes it very easy to move around.

• This depends on the number of independent kinematicconstraints.


• Formally we have the notion of a degree of mobility

δm = 3− number of independent kinematic constraints

• This number is also the number of independent fixed or steerablestandard wheels.

• The independence is important.

• Differential drive has two standard wheels, but they are on thesame axis.

– So not independent.

• So δm = 2 for a differential drive robot

– Can alter x and θ just through wheel velocity.

• A bicycle has two independent wheels, so δm = 1

– Can only alter x using wheel velocity.


Steerability and Maneuverability

• Steering has an impact on how the robot moves.

• The degree of steerability δs is then the number of independentsteerable wheels.

• Note that a steerable standard wheel will both reduce the degreeof mobility and increase the degree of steerability.

• The degree of maneuverability is:

δM = δm + δs

• δM tells us how many degrees of freedom a robot can manipulate.

• Two robots with the same δM aren’t necessarily equivalent (seeon).


Common configurations


Holonomy

• Consider a bicycle, it has a δM = 2 yet can position itselfanywhere in the plane.

– Workspace is the set of possible configurations.– Workspace DOF is 3 in the general case.

• A bicycle only has one DOF that it can control directly.

– Differential DOF– DDOF is always equal to δm

• A general inequality:

DDOF ≤ δM ≤ DOF

• A robot with DDOF = DOF is called holonomic


If we were interested

•We could compute forward and reverse kinematics for robotarms, and use these to decide how to rotate motors to move thehand.


Legged robots

• Of course, this is how we get robot legs to do what we want also.


Summary

• This lecture started by looking at locomotion.

• It discussed many of the kinds of motion that robots use, givingexamples.

•We then looked a bit at kinematics

– The business of relating what robots do in the world to whattheir motors need to be told to do.

•We did a little math, but most of the discussion was qualitative.

• The book goes more into the mathematical detail of establishingkinematic constraints.


locomotion & kinematics

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