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CMPUT 412Motion Control – Wheeled robots

Csaba SzepesváriUniversity of Alberta

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Motion Control (wheeled robots)

Requirements Kinematic/dynamic model of the robot Model of the interaction between the

wheel and the ground Definition of required motion

speed control, position control

Control law that satisfies the requirements

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Mobile Robot Kinematics Kinematics

Actuator motion effector motion What happens to the pose when I change the velocity of

wheels/joints? Neglects mass, does not consider forces (dynamics)

Aim Description of mechanical behavior for design and control Mobile robots vs. manipulators

Manipulators: 9 f: £! X, x = f(µ) Mobile robots:

(#1 challenge!)

Physics: Wheel motion/constraints robot motion

x(T) = x(0) +RT

0 _x(t)dt; _x = f ( _µ)

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Kinematics Model Goal:

robot speed Wheel speeds Steering angles and speeds Geometric parameters

forward kinematics

Inverse kinematics

Why not?

),,,,, ( 111 mmnfy

x

Tyx

ii

i

),, ( 111 yxfT

mmn

),,, ( 11 mnfy

x

yI

x I

s(t)

v(t)

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Initial frame: Robot frame:

Robot position:

Mapping between the two frames

Example: Robot aligned with YI

Representing Robot Positions

TI yx

II YX ,

RR YX ,

100

0cossin

0sincos

R

TIR yxRR

Y R

XR

YI

X I

P

YR

XR

YI

X I

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Example

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Differential Drive Robot: Kinematics

wheel’s spinning speed translation of P

Spinning speed rotation around P

P

YR

XR

YI

X I

_xR ;i = 12r _Ái ; i = 1;2;

_yR = 0

_µR ;i = (¡ 1)i+1 r _Ái2l ; i = 1;2

wheel 1

wheel 2

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Differential Drive Robot: Kinematics II.

P

YR

XR

YI

X I

_xr i = 12r _Ái ; i = 1;2;

_yr = 0

! i = (¡ 1)i+1 r _Ái2l ; i = 1;2

! i = R(µ)¡ 1

0

BB@

r2

³_Á1 + _Á2

´

0r2 l

³_Á1 ¡ _Á2

´

1

CCA

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Wheel Kinematic Constraints: Assumptions

Movement on a horizontal plane Point contact of the wheels Wheels not deformable Pure rolling

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)

r

v

P

YR

XR

YI

X I

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Wheel Kinematic Constraints: Fixed Standard Wheel

v = r _Á; v? = 0

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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 this sliding constraint reduces to:

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Wheel Kinematic Constraints: Steered Standard Wheel

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Wheel Kinematic Constraints: Castor Wheel

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Wheel Kinematic Constraints: Swedish Wheel

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Wheel Kinematic Constraints: Spherical Wheel

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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

Lateral movement

1

)(

)()(

sf NNs

f

t

tt

0)()( 21 JRJ Is

31

11 )(

)(

sf NNss

fs J

JJ

)( 12 NrrdiagJ

0)()(1 Is RC

31

11 )(

)(

sf NNss

fs C

CC

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Example: Differential Drive Robot

J1(¯s)=J1f

C1(¯s)=C1f

® = ? ¯ = ? Right weel:

® = -¼/2,¯ = ¼ Left wheel:

® = ¼/2, ¯ = 0

0)()( 21 JRJ Is

P

YR

XR

YI

X I

0)()(1 Is RC

0

@1 0 l1 0 ¡ l0 1 0

1

A R(µ) _»I =µ

J 2Á0

¶

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Mobile Robot Maneuverability Maneuverability =

mobility available based on the sliding constraints+

additional freedom contributed by the steering

How many wheels? 3 wheels static stability 3+ wheels synchronization

Degree of maneuverability Degree of mobility Degree of steerability Robots maneuverability

ms

smM

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Degree of Mobility To avoid any lateral slip the motion vector

has to satisfy the following constraints:

Mathematically: must belong to the null space of the matrix

Geometrically this can be shown by the Instantaneous Center of Rotation (ICR)

0)(1 If RC

)()(

1

11

ss

fs C

CC

0)()(1 Iss RC

IR )(

IR )( )(1 sC

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Instantaneous Center of Rotation

Ackermann Steering Bicycle

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Degree of Mobility++

Robot chassis kinematics is a function of the set of independent constraints the greater the rank of , the more constrained is the

mobility

Mathematically

no standard wheels all direction constrained

Examples: Unicycle: One single fixed standard wheel Differential drive: Two fixed standard wheels

wheels on same axle wheels on different axle

)(1 sCrank )(1 sC

)(3)(dim 11 ssm CrankCN 3)(0 1 sCrank 0)(1 sCrank 3)(1 sCrank

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Degree of Steerability Indirect degree of motion

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 :

Examples: one steered wheel: Tricycle two steered wheels: No fixed standard wheel car (Ackermann steering): Nf = 2, Ns=2

common axle

)(1 sss Crank

20 ss

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Robot Maneuverability Degree of Maneuverability

Two robots with same are not necessary equal Example: Differential drive and Tricycle (next slide)

For any robot with the ICR is always constrained to lie on a line

For any robot with the ICR is not constrained an can be set to any point on the plane

The Synchro Drive example:

smM

M

2M

3M

211 smM

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Wheel Configurations: ±M=2

Differential Drive Tricycle

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Basic Types of 3-Wheel Configs

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Synchro Drive

211 smM

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Workspace: Degrees of Freedom 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) = Bicycle: DDOF=1; DOF=3 Omni Drive: DDOF=3; DOF=3

Maneuverability · DOF!

m11 smM

11 smM

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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

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

DOFDDOF m

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Non-Holonomic Systems

Non-holonomic systems traveled distance of each wheel is not sufficient to calculate the

final position of the robot one has also to know how this movement was executed as a

function of time! differential equations are not integrable to the final position

s 1L s 1R

s 2L

s 2R

y I

x I

x 1 , y 1

x2 , y2

s 1

s 2

s1=s2 ; s1R=s2R ; s1L=s2L

but: x1 = x2 ; y1 = y2

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Path / Trajectory Considerations: Omnidirectional Drive

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Path / Trajectory Considerations: Two-Steer

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Beyond Basic Kinematics

Speed, force & mass dynamics Important when:

High speed High/varying mass Limited torque

Friction, slip, skid, .. How to actuate: “motorization” How to get to some goal pose:

“controllability”

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Motion Control (kinematic control)

Objective: follow a trajectory position and/or velocity profiles as function of time.

Motion control is not straightforward: mobile robots are non-holonomic systems!

Most controllers are not considering the dynamics of the system

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Motion Control: Open Loop Control

Trajectory (path) divided in motion segments of clearly defined shape: straight lines and segments of a circle.

Control problem: pre-compute a smooth trajectory

based on line and circle segments Disadvantages:

It is not at all an easy task to pre-compute a feasible trajectory

limitations and constraints of the robots velocities and accelerations

does not adapt or correct the trajectory if dynamical changes of the environment occur

The resulting trajectories are usually not smooth

y I

x I

goal

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y R

x R

goal

v(t)

(t)

starte

Feedback Control Find a control matrix

K, if exists

with kij=k(t,e)

such that the control of v(t) and (t)

drives the error e to zero.

232221

131211

kkk

kkkK

y

x

KeKt

tvR

)(

)(

0)(lim

tet

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Kinematic Position Control: Differential Drive Robot

Kinematics:

: angle between the xR axis of the robots reference frame and the vector connecting the center of the axle of the wheels with the final position

vy

xI

10

0sin

0cos

y

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Coordinate TransformationCoordinate transformation into polar coordinates with origin at goal position:

System description, in the new polar coordinates

y

for for

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Remarks

The coordinates transformation is not defined at x = y = 0;

as in such a point the determinant of the Jacobian matrix of the transformation is not defined, i.e. it is unbounded

For the forward direction of the robot points toward the goal, for it is the backward direction.

By properly defining the forward direction of the robot at its initial configuration, it is always possible to have at t=0. However this does not mean that remains in I1 for all time t. a

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The Control Law It can be shown, that with

the feedback controlled system

will drive the robot to The control signal v has always constant sign,

the direction of movement is kept positive or negative during movement

parking maneuver is performed always in the most natural way and without ever inverting its motion.

000 ,,,,

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Kinematic Position Control: Resulting Path

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Kinematic Position Control: Stability Issue

It can further be shown, that the closed loop control system is locally exponentially stable if

Proof: for small xcosx = 1, sinx = x

and the characteristic polynomial of the matrix A of all roots

have negative real parts.

0 ; 0 ; 0 kkkk

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Summary Kinematics: actuator motion effector motion

Mobil robots: Relating speeds Manipulators: Relating poses

Wheel constraints Rolling constraints Sliding constraints

Maneuverability (DOF) = mobility (=DDOF) + steerability

Zero motion line, instantaneous center of rotation (ICR)

Holonomic vs. non-holonomic Path, trajectory Feedforward vs. feedback control Kinematic control