1 Why Study Dynamics - Field Robotics Center · 1 Why Study Dynamics ... controlling wheel speeds (differential/skid steering). ... Our Kalman filter used these for the system model.

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Dynamics 1:1

ynamicsd to be both:: consider options whenck the best if possible.

ond to what is happening

y must be both smart and

eoff in algorithmic designlways a compromise.g both smart and fast, twoy:: ab l e t o p r ed i c t t hef candidate actions.

ble to do precisely what itsis necessary.stly about how to measurew and how to predict theur actions. Both involve

zo Kelly February 2, 2005

• responsive: atold when this

• This section is mowhat is going on noconsequences of yodynamics.

Mobile Robot Systems

Dynamics 1:Aspects of Ground Vehicle Dynamics

1 Why Study D• Mobile robots nee

• deliberativethey exist. PiThink ahead.• reactive: respnow.

• In plain terms, thefast.• This is a key tradand the solution is a• In support of beinother attributes appl

• p r ed i c t i veconsequences o

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Dynamics 1:2

2 P•o

last resort response whichto continue moving. There are many:nsistently) recognize ant was too late to deal withlly (e.g. by turning).stacles are blocking the

s not responding preciselyread the needle” and the is no longer feasible.rake because it is time tor change direction.rake just to slow down -ntrolling speed on roughllowing something you

ontrols, the ultimate gameonment to exert forces onovement of actuators areey have no effect on thele.


brakes) you often need to knowwhere you will stop, not where youhit the brakes.

especially when coterrain or when fodon’t want to hit.• As for all vehicle cis to cause the envirthe vehicle. The minternal forces, thmotion of the vehic

Mobile Robot Systems 2 Predictive Modelling

redictive Modelling Generally, in considering the consequencesf actions, many processes must be modelled.

• The propagation of information throughthe system.

• It takes time to understand whatyou are looking at, to decide what todo, and to do it. • Often you are still moving while allof this is going on.

• The physics of executing the chosenaction.

• Often, what you want and what youget are different.• Even after you “do it” (say, hit the

3 Braking• Braking is often aindicates a failure reasons for the failu

• Did not (coobstacle until iit more gracefu• Dynamic obway forward.• The vehicle ienough to “thgoal trajectory

• Sometimes you bcome to a stop and/o• Sometimes you b

Alon

Dynamics 1:3

•ks

•kac

•tf

agram the external forcesvity and friction:

ly that these forces arer vehicle generates more

s take more time to stop?energy formulation of thework done by the extermalial kinetic energy.

braking distance and wethat gravity does no work

f µsFn µsmg= =

µsmgsbrake


• that propulsive forces are removedinstantly;

• and that the vehicle is translating(so that a particle model will suffice).

Assume that the wheels lock instantly andhe vehicle goes instantly in a sliding mode ofriction.

where is thehave used the fact (on level ground).

12---mv

2 =

sbrake

Mobile Robot Systems 3 Braking

Avoiding col l i s ion requi res prec isenowledge of the time and space required totop.

• You have to look for obstaclesbeyond this distance to have a hopeof avoiding them. • This distance depends alot onspeed, terrain material (ice?), andslope, so outdoor robots often needan explict model.

Consider the vehicle to be moving at somenown velocity at the point where the brakesre applied. The real situation is quiteomplicated. To simplify:\

• assume that the brakes are appliedinstantly;

• On a free body diare the forces of gra

• Note particularcoupled. A heaviefriction.• Do heavier vehicle• Lets see. Use an mechanics. Let the forces equal the init

v

Fn mg=

Alon

Dynamics 1:4

•

•t•opr••sf

rk done by the extermalial kinetic energy.

o?ring (2) to (1), the quantityominator is an “effective”n:

e written as:

gcθ mgsθ– )sbrake

v2

2g µscθ sθ–( )----------------------------------- (2)

µscθ sθ–( ) (3)

v2

2µeffg---------------=


θ

mgsθ

mgcθ

f µsFn µsmgcθ= =

mg

• And (2) can then b

sbrake

Mobile Robot Systems 3 Braking

We can solve for the braking distance:

Soooooo.... Do heavier vehicles take moreime to stop? They probably do - maybe this is an aspectf rolling friction depending on groundressure or because braking force does noteally scale with vehicle weight. How does gravity matter? Consider the same situation on a downwardlope. Assume the surface is rigid, so normalorce governs sliding friction:

sbrakev2

2µsg------------= (1)

• Again, let the woforces equal the init

• Solving:

• Where did mass g• Clearly, by compain brakets in the dencoefficient of frictio

12---mv

2 µsm(=

sbrake =

µeff =

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Dynamics 1:5

tic for Slopes

•t

•ww

•d

equate the change in thetic plus Potential) to theternal forces?s change in kinetic energyy external forces. Gravitytoo.eral case of rough terrain,e magnitudes which varyessary to equate the workinetic energy loss:

gh Heuristic for Slopesitical angle, we can maketions and ive” coefficient of frictionlike (1) is:

% (0.1 rads) reduces then by 10%.

s 12---mv2=

cθ 1= sθ θ=

scθ sθ– ) µs θ–≈


hich gravity overcomes friction. This ishen:

For the parameters used above, its around 25egrees.

µscθ sθ– 0 θtan⇒ µs= =

small angle assumpand the new “effectthat makes (2) look

• So, a slope of 10coefficient of frictio

µeff θ( ) µ(=

Mobile Robot Systems 3 Braking3.1Example: A Rough Heuris

So, as the inclination angle increases, we gethe following behavior:

Clearly, there is a critical angle beyond

B r a k in g D is ta n c e v s S lo p ea t V = 2 .0 m /s e c , µ = 0 .5

0

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

4 5

5 0

0 0 .1 0 .2 0 .3 0 .4 0 .5

T e r r a in D o w n s lo p e in R a d s

Bra

king

Dis

tanc

e• Don’t we need tototal energy (Kinework done by all ex• Nope. Physics sayequals work done bis an external force • For the more genall of the forces havwith time, so its necline integral to the k

3.1 Example: A Rou• Well below the cr

F d•0

s

∫

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Dynamics 1:6

•i

•a•

ur wheels, you cause theoments on your vehicle angular velocity. controlling wheel steerkinematic steering) or byspeeds (differential/skid

e effect of the steeringither curvature (articulated velocity (skid steering).ng of mechanics (Chasle’sber right) is that all rigid

e plane can be consideredbout some point - the

r of rotation (ICR).ring, we try to reduce orkid (and the associatedtuating all wheels in orderth a single instantaneous


RULE OF THUMB:

1. Note that when is small .x 1 x–[ ] 1– 1 x+≈

On a smal l s lope of , brakingdistance increases by % brakingdownhi l l and decreases by %braking uphill.

θθ

θ

to be a rotation ainstantaneous cente• In kinematic steeeliminate wheel senergy losses) by acto be consistent wicenter of rotation.

Mobile Robot Systems 3 Braking1.1Kinematic Steering

The ratio of sloped to level braking distances1:

So the braking distance also increases bypproximately 10%.

sθs0----

v2

2µeffg---------------

v2

2µsg------------

---------------µsµeff--------

µsµs θ–--------------= = =

sθs0---- µs

1

1 θµs-----–

-------------- µs 1 θµs-----+ µs≈ ≈=

1 Turning• When you turn yoterrain to exert mwhich will create an• You may turn byangles (articulated/controlling wheel steering).• In most cases, thactuator is to alter esteering) or angular

1.1 Kinematic Steeri• A basic theoremTheorem if I remembody motions in th

Alon

Dynamics 1:7

ering

•J

•daea•fc

1.2•T

derivatives of the steer

changes in the position ofe huge changes in the or velocity of the wheels.to be able to respond welly well to arbitrary changes

g is by far the simplest tolready seen the Ackermanand its associated “bicycle

sddβ

α

α

R

L

V αtanL----------------

(4)


bout any point within the vehicle envelope. If there are limits on steer angles, there is aorbidden region where the instantaneousenter of rotation (ICR) cannot lie. Dynamics in Kinematic Steering Actuators don’t move at infinite speeds.here are certain to be practical limits on the

β sd td κV= = =

Mobile Robot Systems 3 Braking1.2Dynamics in Kinematic Ste

Consider 4 wheel steer in the followingeantaud Diagram:

This vehicle can crab steer - move in anyirection - without changing heading. If therere no limits on steer angles, such schemes arextremely maneuverable - they can even spin

first and second angles.• Worse yet, small the ICR can causrequired steer angleThey are not likely or to respond equallin ICR position.• Kinematic steerinanalyze. We have asteer configuration model”:

κ 1R---

αtanL------------= = =

· dβds

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Dynamics 1:8

ering

•t

•sbon•A

•mf

is small, and in another equation thus:

e of is limited, then theis limited.y back substitution leads

integrate , and to get . If you ask for a signal

α L 1=

α t( )=

α

V t( ) θ t( )( )dtcost

V t( ) θ t( )( )sin dt

V t( )α t( )dt0

t

∫

αθ


These are pretty important equations inobile robots. Our Kalman filter used these

or the system model.

dt------------ V t( ) θ t( )sin=

dθ t( )dt

------------- V t( )κ t( )=

• So, to get , you, you integrate

y t( ) y00

t

∫+=

θx y,( )

Mobile Robot Systems 3 Braking1.2Dynamics in Kinematic Ste

Suppose that the steering wheel angle (andhe wheel “steer angle” ) satisfy:

If you ask for some arbitrary instantaneousteer angle rate, you will get up to this much,ut only after the transient response period isver. What is the impact of limited rate andeglected transients? Lets see. The “kinematics” (really kinetics) of theckermann configuration are:

α

α· α· max≤

dx t( )dt------------ V t( ) θ t( )cos=

dy t( )

• If we assume that(4) then, we can add

• If the rate of changrate of change of • “Solving” these bto:

κ t( )

κ

x t( ) x00∫+=

θ t( ) θ0 +=

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Dynamics 1:9

ynamics

p

•iie

1.3•imIt

α

max steer angle rate of 30 (and a trapezoidal speed).ant to avoid the obstacleot try to avoid the obstaclee right at any speed aboveodelled slow steering

ly cause a collision.

-2 2 6 10 14ordinate in Meters

0.5 m/s1.0 m/s

1.5 m/s2.0 m/s

2.5 m/s3.5 m/s

0.01+ m/st = 0

eed Reverse Turn. Response toard right is illustrated at various


This graph is for a degrees per secondcommand generator• Clearly, if you wshown, you better nby a sharp turn to th3.5 m/s. The unmresponse will actual

Mobile Robot Systems 3 Braking1.3Example - Reverse Turn D

which the system does not executeerfectly, then the error in following it is:

Note that this error is integrated twice beforet becomes an error in position so small errorsn steer angle quickly accumulate into largerrors in position. Example - Reverse Turn Dynamics The worst thing you can ask for is andnstantaneous change in curvature from theaximum positive to the maximum negative.

ntegrating the true dynamics in this “reverseurn” gives the results shown below:

t( )

δα t( ) αcommanded t( ) αresponse t( )–=

-14 -10 -6X co

-10

-6

-2

2

6

10

14

18

Y c

oord

inat

e in

Met

ers

5.0 m/s

9.0 m/s

obstacle

Figure 3 Constant Spa command to turn hoperating speeds.

Alon

Dynamics 1:10

ynamics

•ssg

Tctp•l

of constant curvature arcs

integrated effect, many ofcome significant even at0 mph.


his example is for a speed of 5.0 m/sec. Itlearly takes about 10 meters of motion beforehe steering wheel even reaches the halfwayoint. Many modern robots reason in terms ofinear curva ture po lynomia l s (ca l l ed

Figure 4 - Constant Speed Partial Reverse Turn

Mobile Robot Systems 3 Braking1.3Example - Reverse Turn D

Now, consider the response to differentteering commands at the same speed. Theame dynamic model used in the last graphives:

-20 -16 -12 -8 -4 0 4X coordinate in Meters

-8

-4

0

4

8

12

16

Y c

oord

inat

e in

Met

ers

t = 0- κmax

+κmax0.0

“clothoids”) insteadfor these reasons.• Due to the doublythese issues can bespeeds well below 1

Alon

Dynamics 1:11

2 •rT

2.1•csm

wi•a•k•

m is l inear, i t can be

ck a diagram like this:

n over Rough Terrainxpected of robots these in a 3 dof subspace of the

de (not heading) isterrain contact constraintis also.

x t( ) Bu t( )+

x t( ) Du t( )+

θ φ,( )


nd inputs are going to be. Of t en , you mus t cons ide r ac tua to rinematics. Often you must consider constraints.

• At most speeds edays, vehicles move6 dof world.• That is, the attitudetermined by the and the elevation z

Mobile Robot Systems 2 Motion Prediction2.1Dynamic Models

Motion Prediction In order to predict what is going to happen, aobot must at least know where it is going.his is not nearly as easy as you may think. Dynamic Models In general, the nonlinear state equationsover every kind of kinematic or dynamicystem as a special case so we can certainlyodel a robot like so:

here is the state vector and is thenput. Now, you need to decide what your states

x· t( ) f x t( ) u t( ) t, ,( )= (5)

x t( ) u t( )

• When the systerepresented by:

• Which gives a blo

2.2 Motion Predictio

x· t( ) A=

y t( ) C=

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Dynamics 1:12

ugh Terrain

•co•

•tgimp

is is:

e e r i n gn a m i c s

V e h i c l eK in e m a t ic s

α r e s p o n s e

r o t t len a m i c s

V r e s p o n s e

*

κ

β·

td∫ψψ·

V

td∫

x y zT

T e r r a i nC o n t a c t

θ

φ


oves, instantaneously in the direction it isointed.

C o o r d i n a teT r a n s fo r m

Mobile Robot Systems 2 Motion Prediction2.2Motion Prediction over Ro

The position and heading are notonstrained but we also have no direct controlver them. In 2D, a reasonable dynamic model again is:

We can convert this to 3D by realizing thathese equations apply in the body frame ine ne ra l . Tha t i s , t h e r obo t r o t a t e s ,

nstantaneously, about its vertical and it

x y,( ) ψ

dx t( )dt

------------ V t( ) θ t( )cos=

dy t( )dt------------ V t( ) θ t( )sin=

dθ t( )dt------------- V t( )κ t( )=

• A diagram of all th

D e l a y

α Cα d e l a y e d

S tD y

D e l a y

V CV d e l a y e d

T hD y

C o o r d in a t eT r a n s fo r m

x· y· z·T

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Dynamics 1:13

3 4 S

•rftm

•t

o•w

tse terrain-induced by:n obstacle sidewaysgh wheel over a rock on a

ayload on a forktruck into the doorway.lso be maneuver-induced

sharply for the present

o fas t for the present

quickly on a downslope.arply on a slope.


• high centers of gravity• high inertial forces, and • steep slopes

ccur regularly. These are complicated phenomena which weill model reasonably well.

• turning too sh

Mobile Robot Systems 3 Vehicle Rollover and Tipover4.1Types of Incidents

Vehicle Rollover and TipoverHOW VIDEOS

While it is probably not a concern for homeobots, factory and warehouse robots (e.g.orktrucks) and field robots (e.g. excavators)ypically must pay alot of attention to thisatter. Why?

• Humans tip lift trucks in warehousesvery often.• Humans tip rough terrain vehicles veryoften.

Its simply a fact of physics (and economics)hat some combinations of

• narrow wheel spacing,

4.1 Types of Inciden• An incident may b

• sliding into a• driving the hislope.• running the pthe wall above

• An incident may aby:

• turning toospeed.• dr iv ing tocurvature.• stopping too

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Dynamics 1:14

•

lityuish the point of wheeltentially reversible) from center of gravity (cg) hasne between the remaining

truck Rollover When Turning

ter Rinner

a


4.2 Forms of Instabi• We must distingliftoff (which is pothe point where thetravelled over the li

Figure 6 - Fork

Mobile Robot Systems 4 SHOW VIDEOS4.2Forms of Instability

Some examples are shown below:

Figure 5 - Rover Tipover When Stopping on a Downslope

a

mg

•

Rou

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Dynamics 1:15

ci•tcps

4.3•Cssr

designate “upper” andr this rigid body to be in

ce balances are:

uilibrium, the sum of the any axis must cancel.wheel contact point andd it:

that the s lope s lowlye. Intuitively, the forces oncrease and those on these in order to counteractncy to tip over. force on the upper wheel the precise instant of loss

mg φsin=

mg φcos=

φh mg φ t2---cos=


fzl

fzufyl

fyumg

φ

h continues to increasthe lower wheel inupper wheel decreathe increasing tende• At some point, thegoes to zero. This is

Mobile Robot Systems 4 SHOW VIDEOS4.3Static Case

on tac t whee l s (wh ich i s no rma l lyrreversible). The second case is more serious and hardero predict (requires inertia models). The firstase is more conservative and easier toredict/detect so we will concentrate on thisituation. Static Case For a static vehicle, the physics are easy.onsider a rectangular vehicle on a slope ashown. Assume that the vehicle and itsuspension is rigid, and that the forcesepresent both the front and rear wheels :

t

• The subscripts “lower” wheels. Foequilibrium, the for

• For rotational eqmoments aroundChoose the lower sum moments aroun

• Now, imagine

fyu fyl+

fzu fzl+

fzut mgsin+

Alon

Dynamics 1:16

ob

•i

•vbt•dt

quilibrium, the figure

gests, the condition iswhen the gravity vector,

center of gravity, points attact point.hen you know that gravityat must be considered. Ifrclockwise moment aboute upper wheel will resist

zl

fzu

fyu

h

mg φcos

mg φsinmg

φ


isappeared, this result can be understood inerms of the direction of gravity alone. At the

1. You lose equilibrium because the terrain would have to pull on the upper wheel in order to prevent the vehicle from tipping. The terrain will push back but it will not resist wheel liftoff.

satisfied precisely emanating from thethe lower wheel con• This is obvious wis the only force thgravity has a countethe lower wheel th

Mobile Robot Systems 4 SHOW VIDEOS4.3Static Case

f equilibrium1. At this instant, the momentalance becomes:

Therefore, the slope at which tipping occurss given by:

This is a pretty well-known result in variousehicle manufacturing industries. In part,ecause if you know the slope and the wheelread, you can compute the height of the cg. S i n c e a l l f o r c e s b u t g r a v i t y h a v e

mg φhsin mg φ t2---cos=

φtan t2h------=

point of loss of ebecomes:

• As the inset sug

ffyl

mg

φ

t

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Dynamics 1:17

mp

4.4•ceir•ppbaflw•dp

, the subscript “o” means the subscript “i” means

nts around the outsidet, and using the vehicler expressing components

fzo

fyomg

hmay

gsφh mgcφ t2---+ 0=


The friction forces have been reversed to theirection in which they are likely to beositive in a roll incident.

body frame axes foof forces, we have:

fzit– mayh– m+

Mobile Robot Systems 4 SHOW VIDEOS4.4Dynamic Case

otion, so equilibrium loss happens at theoint of vanishing moment. Dynamic Case T h e d y n a m i c c a s e i s a l i t t l e m o r eomplicated. Let the same vehicle bexecuting a turn on this inclined plane (whichs not a banked turn). Assume that there is nooll acceleration (steady turn). We use D'Alembert's law to convert theroblem to a simpler "equivalent" staticsroblem. The resultant inertial force wille assumed to act parallel to the groundplane,nd hence it is aligned with the vehicle bodyrame y axis. This assumption amounts toocally flat terrain (instead of a banked turnhich is actually not locally flat).

may

• Now in the figureoutside wheel andinside wheel.

• Summing momewheel contact poin

fzi

fyiφ

t

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Dynamics 1:18

•h

•rptttr

an be interpreted in termsncontact force. When the

e wheel contact point, the to lift off.

e is just a special case of

sφφ-------- t

2h------=

ay gsφ–

g a–


points at the outsidinside wheel begins• Note that:

• the static casthis result.

f =

Mobile Robot Systems 4 SHOW VIDEOS4.4Dynamic Case

solving for the lateral acceleration in g’s, weave:

Rollover commences when the verticaleaction on the inside wheel vanishes. At thisoint, groundplane forces cannot counteracthe moment tending to tip the vehicle towardhe outside of the turn. Setting leadso the threshold on lateral acceleration whereollover begins to occur:

ayg-----

t2---cφ hsφ

tfzimg-------–+ h⁄=

fzi 0=

ayg-----

t2---cφ hsφ+ h⁄=

• Rewriting gives:

• Once again, this cof the resultant novector specific force

ay g–gc------------

gcφ

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Dynamics 1:19

4.5•tba

pyramid formed with thes with the cg at the apex.

ot be in the same plane.en adjacent wheels is axis because when a rigidy two adjacent wheels stay

ected counterclockwise as along a potential tipover

a

f


• Each line betwepotential tipover avehicle lifts off, onlin contact.• If is a vector dirviewed from above

a

Mobile Robot Systems 4 SHOW VIDEOS4.5Stability Pyramid

• lowering the cg or widening the wheelt r e ad o r r eve r s ing t he s l ope , o rdecreasing the acceleration (slowingdown or turning less sharply), increasesstability.• the specific force vector points alongthe direction that a pendulum at the cgwould point.• on flat ground, some vehicles cannotrollover during cornering - the wheelswill slip sideways first.

Stability Pyramid Generally, vehicles are not rectangles, buthe theory generalizes to arbitrary shapes. Theasic issue is whether the net noncontact forcecting at the cg, remains within the stability

pyramid. This is thewheel contact point

• The wheels need n

r

Alon

Dynamics 1:20

l

at

ia•

•uo•a•ab

4.6•it

ction is known from the

ce lera t ion should be:peed and curvature will bet’s control.the coefficient of frictionthe deceleration.tions say the margin is tooan choose a more stableng down or turning less

g to avoid an obstacle on a tipover, the robot must

Controln use all this predictivelys its present specific force

ctive measures would bedictive processes wouldiled.btleties though.


This formulation applies to the total inertialcceleration whether it is caused by turning orraking or anything else. Predictive Stability Control A mobile robot can use all this predictivelyf it knows/measures the attitude and shape ofhe upcoming terrain and controls it motion.

• A mobile robot caif it knows/measureand attitude.• Near liftoff, corretaken because prepresumably have fa• There are many su

Mobile Robot Systems 4 SHOW VIDEOS4.6Predictive Stability Contro

xis, and if is any vector from any point onhat axis directed to the cg, then:

s the moment of the specific force about thexis. Further, the quantity

is positive when the vehicle experiences annbalanced moment and it is zero at the pointf liftoff. The remaining angle before liftoff occurs is kind of stability margin.

r

M r f×=

M a•

• The gravity projepredicted attitude.• The iner t ia l acpredictable because

• for turning, sunder the robo• for braking, and slope give

• If all these calculasmall, the robot coption (like slowisharply).• Note that if brakinslope would cause“pick its poison”.

4.7 Reactive Stability

Alon

Dynamics 1:21

•ir

assuming sliding friction, initial velocity - so yous as far ahead in order toas fast.e in braking distance (not

) is roughly equal to the in radians.iction of the results of

ers requires embeddedr even moderate speeds,

ggressive turning.over can be predicted and of the proximity of the at the cg to the boundary

y pyramid.placed computationally at with knowledge of thegeometry, implements aer detector.


• An inclinometer, the cg, combinedstability polygon basic rollover/tipov

Mobile Robot Systems 5 Summary4.7Reactive Stability Control

• If the robot articulates mass, the cgitself must be computed on-line.• For this purpose alone, it is oftenunnecessary to distinguish gravitationfrom acceleration. An inclinometermeasures precisely the specific forcedirection.• We need the specific force at the cg andits highly unlikely we can put all oursensors there. We need to use thetransformations of moving referenceframes in order to process measurements.

The good news is that a rigid vehicle canmplement reactive stability control inelatively little hardware and software.

5 Summary• Braking distance,is quadratic in themust look four timedrive (safely) twice • The relative changthe distance itself(small) terrain slope• Competent predsteering maneuvdynamic models foand especially for a• Turnover and rollmeasured in termsspecific force actingedges of the stabilit

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Dynamics 1:22


Mobile Robot Systems 5 Summary4.7Reactive Stability Control

1 Why Study Dynamics - Field Robotics Center · 1 Why Study Dynamics ... controlling wheel speeds (differential/skid steering). ... Our Kalman filter used these for the system model.

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