ROBOTICS 01PEEQW Basilio Bona DAUIN Politecnico di Torino Mobile - - PowerPoint PPT Presentation

ROBOTICS 01PEEQW

Basilio Bona DAUIN – Politecnico di Torino

Mobile & Service Robotics Kinematics

Fundamental problems in mobile robotics

Locomotion: how the robot moves in the environment Perception: how the robot perceives the environment Mapping: how to build the map of the environment Localization: where is the robot wrt the map Representation: how the robot organizes the knowledge about the environment Path planning/action planning: what the robot shall do to go from here to there; what are the actions to be performed to complete a specified task Supervision and control: how are the command to actuators generated to perform simple or complex tasks. How to generate tasks

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Kinematics

robot kinematics depends on the adopted locomotion stuctures Almost all mobile robots are underactuated, i.e., they have less actuators (and the relative control signals) that is less than the number of degrees of freedom of the structure Various wheels arrangements exist Each one has a different kinematic model We will study the following structures

Differential drive robots Bicycle-like robots And a brief introduction to quadcopters

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Examples of underactuated structures

A car has two actuators, thrust and steer, and moves in a 2D space (3dof) A fixed-wing aircraft has four actuators, forward thrust, ailerons, elevator and rudder, and moves in a 3D space (6dof) A helicopter has four actuators: thrust vector magnitude and direction from the main rotor, and yaw moment from the tail rotor, and moves in a 3D space (6dof) A quadcopter has four actuators, the four rotor thrusts , and moves in a 3D space (6dof) A ship has three actuators: two propellers on parallel axes and

• ne rudder, and moves in a 2D space (3dof)

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Instantaneous Curvature Centre (ICC)

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ICC

If an ICC exists, each wheel rotates without slippage The wheel axes meet in a common point called ICC

ICC

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? wheels slip = skid steering

If an ICC does not exist, the wheel motion occurs with slippage

Kinematics of a wheeled robot (aka rover)

the steering angle is the angle between the velocity vector and the steered wheel direction is equivalent to the angle between the normal to the velocity vector and the steered wheel rotation axis the robot pose is the position and the orientation (with respect to some give inertial axis)

( )

( ) cos sin sin cos 1

m

t x y θ θ θ θ θ =   −        =             p R

T

( ) t β

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β θ

Kinematics

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d

m

ω α α δ rϕ = v ɺ

W

ω

m

i

m

j

W

i

W

j

R

m

R x y

W

R

Kinematics – Fixed wheel

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The kinematic model represents a rover frame with a generic active non- steering wheel, located at a given position, with a local orientation The kinematic equations describe the relations and constraints between the wheel angular velocity and the angular velocity of the frame

( )

x y

v v = v

T

( )

W Wx Wy

ω ω = ω

T

( )

m mz

ω = ω

T

Linear velocity of the wheel at the contact point with the plane Angular velocity of the wheel Angular velocity of the robot

, α δ

( )

cos sin d d α α = d

T

given constants

Kinematic constraints

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Hypothesis

m W

r θ ϕ ϕ = = = v ɺ ɺ ɺ ω ω

Constraints: To avoid slippage the tangential velocity at the wheel contact point, due to the rover rotation around its center, must be equal to the advance velocity of the wheel

In the wheel reference frame we have

( ) ( ) ( )

T T T T

1 1 1

m W

r θ ϕ ϕ = = = v ɺ ɺ ɺ ω ω

Differential Drive (DD) Rover

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Left wheel, active, non steering Passive support (castor, omniwheel) Right wheel, active, non steering

Differential Drive Rover

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m

i

m

j

m

k

Left wheel

2 ℓ

r

Right wheel

2 ℓ

Differential Drive Kinematics

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( )

t θ ɺ

( )

t v

( )

t vℓ

( )

r t

v

ICC

R

m

R

Differential Drive Kinematics

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( )

t vℓ

( )

r t

v

( )

t θ ɺ

( )

t v

ICC

R

Differential Drive Kinematics

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vℓ

r

v v

ICC

i

C

i

ρ

Consider the generic time instant

ℓ

Differential Drive Kinematics

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m

R

before after

vℓ

r

v v vℓ

r

v v

i

C

r >

v vℓ

r

> v v

ℓ

Turn left Turn right

m

R

t δθ θ

Differential Drive Kinematics

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k

s A B AB = t

• 2

k

δθ

ik

C

k

δθ

rk

s

k

sℓ

Differential Drive Kinematics

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ik

C

1 k+

t

k

θ

k

δθ i j

rk

t

ik

c

ik rk

ρ − j

ik rk

ρ j

R

k

θ

k

δθ

k

v

k

v

Differential Drive Kinematics

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Differential Drive Kinematics

Considering the sampled time equations and assuming constant values during the time interval

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( / 2) ( / 2)

k k rk k k k

v v ω ρ ω ρ + = − =

ℓ

ℓ ℓ ICC sin cos

xk k k k k yk k k k

C x C y ρ θ ρ θ     −     = =     +         ; 2 ( ) 2 ( )

rk k rk k k k rk k k k rk k k k rk k

v v s s v v v v v v v v ω δθ ρ ω − − = = + = + = = −

ℓ ℓ ℓ ℓ ℓ

ℓ ℓ ℓ

Differential Drive Kinematics

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1

d d

k k k k k

T t ω θ ω θ θ δθ

+

= ⇒ − ≃ ≐

1 1 1

cos( ) sin( ) sin( ) cos( ) 1

k k k k xk xk k k k k yk yk k k k

x x C C y y C C δθ δθ δθ δθ θ θ δθ

+ + +

        − −                 = − +                                

Approximation when sampling period is small

( ) cos ( ) ( ) d ; ( ) sin ( ) ( ) d ; ( ) ( ) d

t t t

x t v y t v t θ τ τ τ θ τ τ τ θ ω τ τ = = =

∫ ∫ ∫

(A)

Differential Drive Kinematics

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1 1 1

cos sin

k k k k k k k k k k k

x x v T y y v T T θ θ θ θ ω

+ + +

= + = + = +

EULER APPROXIMATION

Differential Drive Kinematics

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1 1 1

1 cos 2 1 sin 2

k k k k k k k k k k k k k

x x v T T y y v T T T θ ω θ ω θ θ ω

+ + +

    = + +            = + +        = +

RUNGE-KUTTA APPROXIMATION

Differential Drive Kinematics

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EXACT INTEGRATION

( ) ( )

1 1 1 1 1

sin sin cos cos

k k k k k k k k k k k k k k k

x x v y y v T ω θ θ ω θ θ θ θ ω

+ + + + +

= + − = − − = +

(B) (A) and (B) are the same

Differential Drive Kinematics

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Path Planning

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Path Planning

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initial pose final pose initial pose final pose

• bstacles

free space

2D

Path Planning

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3D F I

Path Planning A robot path from pose A to pose B can always be decomposed into a series

• f arcs of different radius

arcs of zero radius represent turn-in-place maneuvers arcs of infinite radius represents straight line maneuvers

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Path Planning

Driving a vehicle to a goal (no orientation specified)

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θ − (0,0) ,

f f

x y (0, ) ρ −

A

R

A

relative to frame R

Path Planning

Driving a vehicle to a goal (orientation specified)

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1

θ − (0,0)

1 1

,

f f

x y

1

(0, ) ρ −

A

R

2

θ

2

(0, )

y

r

B

R 2 2

,

f f

x y

A

relative to frame R

B

R

Path Planning

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Path Planning

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Unicycle

Unicycle is the simplest example of a wheeled robot. It consists of a single actuated steering wheel ω is the steering velocity

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m

R R

x y

θ

r

v cos sin 1 x y v θ θ ω θ                   = +                         ɺ ɺ ɺ

control commands (underactuated)

x y θ      

T

states/coordinates

Unicycle motion

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Unicycle-like robot (polar coordinates)

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x

θ δ

r

R

r

v

ρ α y

2 2

( , ) ( , ) x y y x y x ρ α θ δ θ α = + = − = + = atan2 atan2

θ′

Bicycle-like robot

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x y

r

R R θ L φ v cos sin tan x v y v v L θ θ θ φ = = = ɺ ɺ ɺ

NON HOLONOMIC CONSTRAINT

sin cos x y θ θ − = ɺ ɺ Lθ ɺ

f

y

f

x

steering angle

• rientation

Bicycle equations 1

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1’) 2) 1)

Bicycle equations 2

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Bicycle equations 3

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Bicycle equations 5

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Bicycle vs unicycle motion

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Quadcopters

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Quadcopter model

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ℓ

4

F

B

R

3

N

B

k

B

i

B

j

4

N

2

F

2

N

3

F

1

N

1

F

Quadcopter equations

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Quadcopter equations

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Quadcopter equations

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Quadcopter equations

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Odometry

Odometry is the estimation of the successive robot poses based on the wheel motion. Wheel angles are measured and used for pose computation

Odometric errors increase with the distance covered, and are due to many causes, e.g.

Imperfect knowledge of the wheels geometry Unknown contact points: in the ideal DD robot there are two geometric contact points between wheels and ground. In real robots the wheels are several centimeters wide and the actual contact points are undefined Slippage of the wheel wrt the terrain

Errors can be compensated sensing the environment around the robot and comparing it with known data (maps, etc.)

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Odometry Errors

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Map Desired path Estimated path based only on odometry

Visual odometry

Visual odometry is the estimation of the robot poses based only on the acquisition of successive images of the environment Stereo visual odometry uses two cameras to estimate the ego- motion of a camera by tracking features in a video sequence. If the camera system is calibrated then the full 3D Euclidean geometry can be reconstructed Using a single camera, the reconstruction is up to an unknown scale factor A RGB-D (Kinect or similar) camera can be used

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Other important problems that must be addressed

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• 1. Mapping: how to build environment maps (geometrical vs

semantic maps)

• 2. Localization: where am I in the map?
• 3. Simultaneous Localization and Mapping (SLAM): build a map

and at the same time localize the robot in the map while moving

• 4. Path planning: the definition of an “optimal” geometric path

in the environment from A to B, and avoid obstacles

Path Planning – A map is given

Path planning algorithm: it computes which route to take, from the initial to the final pose, based on the current internal representation of the terrain possibly considering a “cost function” (minimum time, minimum energy, maximum comfort, etc.) Complete path generation: the path is created as a sequence of successive moves from the initial pose to the final one Motion planning: is the execution of the above defined theoretical route; it translates the plan from the internal representation to the physical movement of the wheels Next move selection: at each step, the algorithm must decide which way to move next. An efficient dynamic implementation is required

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Path Planning

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Introduction to AI Robotics (MIT Press) Robin Murphy 2000

Navigation

• Where am I going? Mission

planning

• What’s the best way there?

Path planning

• Where have I been? Map

making

• Where am I? Localization

Mission Planner Carto- grapher Behaviors Behaviors Behaviors Behaviors

deliberative reactive

How am I going to get there?

Path Planning

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A B

Fixed

• bstacles

Mobile obstacles Acceleration constraints Velocity constraints

Path Planning

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i j k k = 1 k = 2 k = k N =

Which commands shall I give to the rover?

R

Path Planning

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Initial pose Final pose Path A Path B A B

Motion Planning

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Initial pose Final pose Move 1: rotation Move 3:rotation A B Move 2: go straight

Non-holonomic constraints

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Non-holonomic constraints limit the possible incremental movements in the configuration space of the robot Robots with differential drive move on a circular trajectory and cannot move sideways Omni-wheel robots can move sideways

Holonomic vs. Non-Holonomic

Non-holonomic constraints reduce the control space with respect to the current configuration (e.g., moving sideways is impossible) Holonomic constraints reduce the configuration space

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Academic year 2010/11 by Prof. Alessandro De Luca

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Academic year 2010/11 by Prof. Alessandro De Luca

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Academic year 2010/11 by Prof. Alessandro De Luca

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Academic year 2010/11 by Prof. Alessandro De Luca

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