L ECTURE 5 ( CONTINUED ): K INEMATIC CONSTRAINTS D EGREES OF FREEDOM - - PowerPoint PPT Presentation

16-311-Q INTRODUCTION TO ROBOTICS

LECTURE 5 (CONTINUED): KINEMATIC CONSTRAINTS DEGREES OF FREEDOM/MOBILITY

INSTRUCTOR: GIANNI A. DI CARO

2

M O B I L E R O B O T M A N E U V E R A B I L I T Y

Mathematically, the degree of maneuverability 𝜺M is defined as the sum of: Degree of mobility, 𝜺𝙣 and Degree of steerability, 𝜺𝘵 𝜺M = 𝜺𝙣 + 𝜺𝘵 The kinematic mobility (maneuverability) of a robot chassis is its ability to directly move in the environment, which is the result of:

• 2. The additional freedom contributed by steering and spinning steerable wheels

Determine the Degree of steerability, 𝜺𝘵: degrees of controllable freedom based on steering wheels and then moving

• 1. The rule that every standard wheel must satisfy its no sliding and rolling

constraints (↔ Each wheel imposes zero or more constraints on the motion) Determine the Degree of mobility, 𝜺𝙣: degrees of controllable freedom based on changes in wheels’ velocity

3

M A N E U V E R A B I L I T Y A N D ( N O N ) H O L O N O M Y

The degree of mobility quantifies the controllable degrees of freedom

• f a mobile robot based on the changes applied to wheel velocities

Instantaneous center of rotation (ICR) / Instantaneous center of curvature (ICC)

Holonomic: If the controllable degrees of freedom is equal to total degrees of freedom, then the robot is said to be Holonomic, Non-Holonomic otherwise

• 1. In the following we will first derive the controllable degrees of freedom using

algebraic reasoning (matrix rank, notion of linear independence)

• 2. Then, we will show that kinematic constraints of a robot with respect to the

degree of mobility can be also demonstrated geometrically using:

• 3. Finally, we will relate maneuverability, DOFs, holonomic and non

holonomic constraints, fully and under actuated robot systems

4

R E C A P : T Y P E S O F W H E E L S

Steered standard / Orientable Fixed standard Castor / Off-centered orientable

d

Mecanum/Swedish Spherical

5

F I X E D S TA N D A R D W H E E L

˙ ξR = R(θ) ˙ ξI = R(θ)     ˙ x ˙ y ˙ θ    

Reference wheel point A (on the axle) is in polar coordinates: A(l, α)

β: angle of wheel plane wrt chassis

• Rolling constraint (pure rolling at the contact point): All motion along the

direction of the wheel plane is determined by wheel spin

• Sliding constraint: The component of the wheel’s motion orthogonal to the

wheel plane must be zero

P

˙ ϕ v

W h e e l p l a n e Wheel axle

YR XR

A

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F I X E D S TA N D A R D W H E E L : C O N S T R A I N T S E Q U AT I O N S

The wheel, of radius r, spins over time such that its rotational position around the horizontal axle is a function of time: and linear velocity is r

Rolling constraint: projections of Sliding constraint:

𝜷, β, l are parameters in the local {R} frame

along wheel plane must equal linear velocity from wheel projections of

• rthogonal to the wheel

plane must be zero

3 component projection vectors

˙ ϕ ˙ ϕ

R[˙

x ˙ y ˙ θ]

R[˙

x ˙ y ˙ θ]

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FIXED STANDARD WHEEL: ROLLING CONSTRAINT

90-(𝜷+𝜸)

• R[ x

. y . 𝝸 . ] are the components of 𝜊 .

R the pose velocity

vector in the coordinate frame {R} fixed to the robot in the reference point P.

• Projections of all robot’s velocities (linear and

angular) on the wheel’s velocity plane must equal the velocity implied by the wheel’s spinning (under pure rolling assumption)

• Projection of x

. along wheel velocity plane: x . cos(90-(𝜷+𝜸)) → x . sin(𝜷+𝜸)

• Projection of y

. along wheel velocity plane: y . (-cos(𝜷+𝜸))

𝜷+𝜸

• Projection of the robot angular velocity 𝛊

. (-l) along wheel velocity plane: 𝛊 . (-l) cos(𝜸)

FIXED STANDARD WHEEL: SUMMARY OF CONSTRAINTS

Standard wheel A: ⇥ P ˙

xR

P ˙

yR

P ˙

θR

⇤ 2 4 ˙ xR ˙ yR ˙ θR 3 5 = r ˙ ϕ

P ˙

xR, P ˙ yR, P ˙ θR = Projections of ˙

xR, ˙ yR, ˙ θR along the wheel plane

(Pure) Rolling constraint No sliding (no side motion) constraint

Standard wheel A: h P ⊥

˙ xR

P ⊥

˙ yR

P ⊥

˙ θR

i 2 4 ˙ xR ˙ yR ˙ θR 3 5 = 0

P ⊥

˙ xR, P ⊥ ˙ yR, P ⊥ ˙ θR = Projections of ˙

xR, ˙ yR, ˙ θR orthogonal to the wheel plane

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N U M E R I C E X A M P L E : N O S I D E M O T I O N C O N S T R A I N T

⇥ 1 ⇤ 2 6 6 4 1 1 1 3 7 7 5 2 6 6 4 ˙ x ˙ y ˙ θ 3 7 7 5 = ⇥ 1 ⇤ 2 6 6 4 ˙ x ˙ y ˙ θ 3 7 7 5 = 0

α = 0, β = 0, θ = 0 Wheel A is in position such that:

XR

Y

R

XI

YI

A v

.

• Projection of robot’s velocity component along the YR axis on the wheel’s plane

(perpendicular to v) is 0, being perpendicular to each other

• Projection of robot’s velocity component along the XR axis on the wheel’s plane is

1, being wheel’s plane and XR fully aligned

• Projection of robot’s velocity component from rotation d𝜄/dt is 0, because of 𝛾=0
• R(𝜄) = I because of no relative rotation between {R} and {I}

Wheel plane

No instantaneous motion is possible along the x axis of the robot frame, dx/dt=0

10

S TA N D A R D S T E E R A B L E W H E E L

Additional degree of freedom: Now the angle β of the wheel wrt the chassis is not fixed anymore, but is time-varying, as result of the control actions

Instantaneously, the rate of variation of β(t) does not have an impact on the motion

• constraints. It’s only by time integration that the changes in the steering angle have

an impact on robot mobility.

11

C A S T O R W H E E L

✦ The rolling constraint is identical to the standard steering case since the offset axis

does not play a role during motion

✦ The sliding constraint changes: the lateral force on the wheel occurs at A because

this is the attachment of the wheel to the chassis

✦ All lateral motion is balanced by an equivalent caster steering motion such

that no resistance to motion is opposed by the castor wheel

12

C A S T O R W H E E L

✦ In a steered standard wheel, the steering

action does not by itself cause a movement

• f the robot chassis.

✦ In a castor wheel the steering action itself

moves the robot chassis because of the

• ffset between the ground contact point

and the vertical axis of rotation.

Given any robot chassis motion ˙ ξI, there exists some value for spin speed, ˙ ϕ, and steering speed, ˙ β, such that the constraints are met.

Kinematic constraint equations are integrable! (steering must be free)

Therefore, a robot with only castor wheels can move with any velocity in the space of possible robot motions: omnidirectional system

13

S W E D I S H W H E E L

✦ A fixed standard wheel with rollers attached to the wheel perimeter, with

axes that are antiparallel to the main axis of the fixed wheel component.

✦ The exact angle 𝜹 between the roller axes and the main axis can vary

Capable of omnidirectional motion

14

S P H E R I C A L W H E E L

✦ Same equations as in the fixed case, but now β gives the direction of motion,

and is a free variable derived from the sliding constraint equation.

✦ Example: If the robot translates in direction YR, then the second equation

becomes: sin(α+β) = 0, making β = -α Omnidirectional motion. Neither rolling nor sliding constraints

15

A R O B O T W I T H N S TA N D A R D W H E E L S O U T O F M W H E E L S

• A. Each wheel can impose zero or more constraints on the motion
• B. Only fixed and steerable standard wheels impose kinematic constraints

Let’s assume to have N = Nf + Ns standard (Fixed + Steerable) wheels

• Rolling constraints:

J1(βs)R(θ) ˙ ξI − J2 ˙ ϕ = 0 ϕ(t) = " ϕf (t) ϕs(t) # , J1(βs) = " J1f J1s(βs) # , J2 = diag(r1, r2, . . . , rN)

• Lateral movement:

C1(βs)R(θ) ˙ ξI = 0, C1(βs) = " C1f C1s(βs) #

Rolling constraints Sliding constraints

16

A R O B O T W I T H N S TA N D A R D W H E E L S O U T O F M W H E E L S

Sliding constraints

• Rolling constraints:

J1(βs)R(θ) ˙ ξI − J2 ˙ ϕ = 0 ϕ(t) = " ϕf (t) ϕs(t) # , J1(βs) = " J1f J1s(βs) # , J2 = diag(r1, r2, . . . , rN) ϕ(t) = (Nf + Ns) × 1, J1 = (Nf + Ns) × 3

Rolling constraints

J1 is a matrix with projections for all wheels to their motions along their individual wheel planes, which have a fixed angle for the fixed wheels and a time-varying angle for the steerable wheels

• C1(βs)R(θ) ˙

ξI = 0, C1(βs) = " C1f C1s(βs) # C1 = (Nf + Ns) × 3

C1 is a matrix with projections for all wheels to their motions in the direction orthogonal to wheels’ planes

17

D I F F E R E N T I A L D R I V E R O B O T E X A M P L E

XR Y

R

P AR AL l l

chassis

Right wheel: AR ( α = -𝛒/2), β = 𝛒 Left wheel: AL ( α = 𝛒/2), β = 0

l, l,

" J1(βs) C1(βs) # R(θ) ˙ ξI = " J2 ˙ ϕ # " J1f C1f # R(θ) ˙ ξI = " J2 ˙ ϕ #

➔

No steering wheels

18

D I F F E R E N T I A L D R I V E R O B O T E X A M P L E

" J1f C1f # R(θ) ˙ ξI = " J2 ˙ ϕ #

2 6 6 4 " 1 ` 1 −` # ⇥ 1 ⇤ 3 7 7 5 R(✓) ˙ ⇠I = " J2 ˙ ' # = 2 6 6 4 r ˙ '1 r ˙ '2 3 7 7 5

Expliciting wrt the velocity pose of the robot in {I}: kinematic equations for differential drive robot ˙ ⇠I = R(✓)−1 2 6 6 4 1 ` 1 −` 1 3 7 7 5

−1 "

J2 ˙ ' # = R(✓)−1 2 6 6 4 1/2 1/2 1 1/2` −1/2` 3 7 7 5 " J2 ˙ ' #

𝜸=0, wheel plane is aligned with robot Y axis Wheel’s motion is all projected along robot’s X axis Wheel plane, no- motion line, is fully aligned with robot Y axis Linear velocities from wheels 1 and 2 The two no-sliding constraints are equal because the two wheels are parallel, one is redundant, removed

⇥ 1 ⇤

⨉