Robotics (Kinematics)

Winter 1393

Bonab University

Kinematics: most basic study of how mechanical systems behave

• Need to understand the mechanical behavior for:• Design

• Control

Both: Manipulators, Mobile Robots

Manipulator robots: more matured vs. Mobile robots who are following

• Mobile robot community asks similar questions, Example:

• Workspace• Manipulator: range of possible positions achieved by its end effector relative to its fixture

to the environment

• Mobile robot: range of possible poses achieved in its environment

• Controllability• Manipulator: (manner) active engagement of motors used to move from pose to pose in the

workspace

• Mobile robot: defines possible paths and trajectories in its workspace

2

Introduction

Kinematics: how mechanical systems behave

• Dynamics:• Manipulator: places additional constraints on workspace and trajectory due to mass and

force considerations

• Mobile robot: limited by dynamics; for instance, a high center of gravity limits the practical turning radius of a fast, car-like robot

• What is the main difference?

• Position estimation:• Manipulator: one end fixed understanding kinematics of the robot & measuring the

position of all intermediate joints always computable by current sensor data

• Mobile robot: can wholly move no direct way to measure position instantaneously Instead: integrate the motion of the robot over time (Add to this the inaccuracies of estimation due to slippage) (precise) extremely challenging

• understanding the motions of a robot:1. Each wheel’s contribution (enabling to move)

2. Also imposing a constraint (e.g. refusing to skid laterally)

3

Introduction

Mobile Robot Kinematics

This chapter overview:

1. Notation: allowing expression of robot motion in a global reference frame as well as the robot’s local reference frame.

2. Demonstrate construction of simple forward kinematic models of motion (how robot as a whole moves as a function of • Its geometry

• Individual wheel behavior

3. Formally describe the kinematic constraints of individual wheels combinethese kinematic constraints to express the whole robot’s kinematic constraints

4. With these tools: evaluate the paths and trajectories that define the robot’s maneuverability

4

Introduction

Mobile Robot Kinematics

• Kinematics’ Benefits / Usage

• Predict the movement

• Wheel Odometry (use of data from motion sensor to estimate

position change over time)

• Find the distribution of speed and steering

• Design controller & path planner

5

Kinematics

Kinematic Models and Constraints

• Deriving a model for the whole robot’s motion: a bottom-up process:

• Each individual wheel contributes:• motion

• At the same time, imposes constraints on robot motion

• Wheels are tied together (chassis geometry) their constraints combine form: constraints on the overall motion of the robot chassis

• Forces and constraints of each wheel must be expressed with respect to a clear and consistent reference frame

• Also needed: a clear mapping between global and local frames of reference

• Representing robot position:Assumption:

• robot = a rigid body on wheels, operating on a horizontal plane

• The total dimensionality of this robot chassis on the plane =3• 2: for position in the plane

• 1: for orientation along the vertical axis

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Kinematics

Representing robot position

• Of course, there are additional DoF & flexibility

(wheel axles, wheel steering joints, and wheel castor joints),

but by robot chassis = only to the rigid body of the robot, ignoring the joints and DoF internal to the robot and its wheels

• Global reference frame of the plane v.s. local reference frame of the robot?

• Arbitrary: Origin (O)

• Arbitrary: XI, YI (Inertial basis)

• Choose a point P on the robot chassis as its

position reference point XR, YR defines 2 axes

relative to P on the robot chassis

(robot’s local reference)

• Pose of a Robot:

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Kinematics

O x

y

Rotation Matrix connects the frames of reference ( ξI, ξR)

• ξR =Known ξI =?

• Pose change is known in Local frame global?

• ΔXI = ΔXR cos θ - ΔYR sin θ

• ΔYI = ΔXR sin θ + ΔYR cos θ

• ΔθI = ΔθR

• ξI =

ΔXIΔt

ΔYIΔt

ΔθIΔt

= cos θ − sin θ 0sin θ cos θ 00 0 1

ΔXRΔt

ΔYRΔt

ΔθRΔt

• ξI = R-1(θ) ξR

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Kinematics

ΔXI

ΔYI

Δθ

Describing robot motion in terms of component motions

• First: ξI =Known ξR =? (what is the command in robot’s language)

• It’s necessary: to map motion along the axes of the global reference frame to motion along the axes of the robot’s local reference frame

• Of course, the mapping is a function of the current pose of the robot

• Mapping needs : orthogonal rotation matrix:

R(θ) ξI

• The operation is denoted:

• Computation of this operation

depends on the value of θ

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Kinematics

The mobile robot aligned with a

global axis

Describing robot motion in terms of component motions

10

Kinematics

For example:

Θ = 90

• Given some velocity in the global-ref:

• Robot experiences what velocities?

• How about a bit afterwards?

Forward Kinematics:How does the robot move, given its geometry and the speeds of its wheels?

• P centered between the 2 wheels

• Diff. drive robot has two wheels, each with diameter r

• Each wheel is a distance l from P

• Spinning speed of each wheel, 𝜑1 , 𝜑2

• Forward kinematic predicts robot’s overall speed in

the global reference: ξI = R-1(θ) ξR

Forward Kinematics

• How to find f ?

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Kinematics

Forward Kinematics

• How to find f ?

• First: compute the contribution of each of the two wheels in the local

reference: ξR

• Our example of diff. drive robot:

• Contribution 1 wheel’s spinning speed to the translation speed

at P in the direction of +XR (P is half way)

• The other wheel stationary

𝑥𝑟1 = 1 2 𝑟 𝜑1

• In the same way, wheel-2:

𝑥𝑟2 = 1 2 𝑟 𝜑2

• In a diff. drive these 2 components can be simply added to form 𝑋𝑅, of ξR

• Assume two wheels spinning with the same speed in opposite directions 𝑋𝑅 = ?

• 𝑌𝑅 is even simpler to calculate, neither wheel can contribute to sideways 𝑌𝑅 = 0

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Kinematics

Forward Kinematics

• Finally, we have to compute: θ𝑅 , rotational component of ξR

• What is contribution of each wheel?

• Add them

• Consider right wheel (wheel-1) moves forward Robot rotates CCW

Pivoting around Wheel-2 rotation velocity ω1

• ω1 = 𝑟 𝜑

1

2𝑙

• The same calculation for Wheel-2, but its forward movement CW

• ω2 = -𝑟 𝜑

2

2𝑙

• Combining all above:

• Example:

• Robot position: θ=90

• r=1, l=1

• Wheels are unevenly engaged: 𝜑1=4, 𝜑2=2 interpretation

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Kinematics

2l

cos θ − sin θ 0sin θ cos θ 00 0 1

Wheel kinematic constraints

• Previous approach:• Provided motion of a robot given its component wheel

• However, we wish to determine the space of possible motions for each robot chassis design

must go further: describing the constraints on robot motion imposed by each wheel

• Simplifying Assumptions:• Wheels remain vertical

• Single point of contact to ground

• No sliding at this single point of contact

• Under these assumptions: 2 constraints for every wheel type1. rolling contact the wheel must roll when motion takes place in the appropriate

direction.

2. No lateral slippage the wheel must not slide orthogonal to the wheel plane.

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Kinematics

Wheel kinematic constraints: 1-Fixed standard wheel

• No vertical axis of rotation for steering • angle to the chassis = fixed

• it is limited to motion back and forth

• Along the wheel plane

• Rotation around its contact point with the ground plane

• Wheel = A

• Position: polar coordinates by distance l & angle α

• Angle of wheel plane relative to the chassis: β=fixed

• With radius= r, can spin over time: 𝜑(t)

• rolling constraint all motion along the direction of the wheel plane must be accompanied by the appropriate amount of wheel spin pure rolling at the contact point:

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Kinematics

Wheel kinematic constraints: 1-Fixed standard wheel

• Necessary to transform from I to R frame: because all other parameters in the equation: l, α, β are in terms of local reference frame

• In the same way (along the green line):

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Kinematics

+θO

R

+xO

R

Wheel kinematic constraints: 1-Fixed standard wheel

• What is the meaning of those equations?

• Example:

• Wheel is in a position that:

• 2nd equation?

• Further assume that local & global frames

Are aligned

• meaning: constrains the component of motion along XI to be zero and since XI & XR are parallel in this example, the wheel is constrained from sliding sideways, as expected

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Kinematics

Wheel kinematic constraints: 2-Steered standard wheel

• Differs from the fixed standard wheel:

only an additional DoF

• Equations are exactly the same• 1 exception:

• β β(t)

• Constraints are identical to those of the fixed standard wheel because• 𝜑 have a direct impact on the instantaneous motion constraints of a robot

• β does not,

• only by integrating over time that changes in steering angle can affect the mobility of a vehicle

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Kinematics

Wheel kinematic constraints: 3-Castor wheel

• Able to steer around a vertical axis

• This axis of rotation does not pass through

ground contact

• Like steered standard wheel, the castor

wheel has 2 time-varying parameters:• 𝜑(t)

• β(t): steering angle and orientation of AB

• Rolling constraint?

• Movement along XR ( 𝑋𝑅) contribution along wheel plane?

AA

• Movement along YR ( 𝑌𝑅) ? BB

• Movement along θR ( θ𝑅) ? CC

[AA BB CC] x

𝑋𝑅 𝑌𝑅 θ𝑅

=all contributions along the wheel plane=Wheel must roll this amount

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Kinematics

-90

+θO

R

Wheel kinematic constraints: 3-Castor wheel

• AA: Cos((α+β)-90)=Cos(90-(α+β))

= Sin(α+β)

• BB: Cos(180-(α+β))= -Cos(α+β)

• CC: l x Cos(180-β) = -l Cos(β)

• Rolling constraint as before (offset axis plays

no role in motion aligned with the wheel plane)

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Kinematics

-90

+θO

R

180-(α+β)

Wheel kinematic constraints: 3-Castor wheel (sliding constraint )

• This wheel: significant impact on the sliding

constraint (lateral force on the wheel occurs

at point A)

• Lateral movement being zero is wrong

• Instead it’s more like a rolling constraint:

• Motions orthogonal to wheel plane must be balanced by equivalent and opposite castor steering

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Kinematics

+θO

R

[AA’ BB’ CC’] x

𝑋𝑅 𝑌𝑅 θ𝑅

=all contributions perpendicular to the wheel plane=Wheel must steer minus this amount

Wheel kinematic constraints: 3-Castor wheel

• First assume β (steering) is locked &

Find the lateral motion at ground contact

• AA’: Cos(180-(α+β))= -Cos(α+β) (direction)

• BB’: Cos(90-(180-(α+β))= Sin (180-(α+β))

= Sin(α+β)

• CC’: (How much move will rotation cause at

Contact point?): l Cos(β-90)=l Sin(β) (at A)

l Sin(β)+d (at B)

• Now unlocked steering should

Compensate this lateral skid with arm length = d

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Kinematics

+θO

R

+θO

R

180-(α+β)

Wheel kinematic constraints: 3-Castor wheel

• Last result is critical to the success of castor wheels • because by setting the value of β

𝑜(t) any arbitrary lateral motion can be acceptable.

• In steered standard wheel, steering action → movement chassis.

• In castor wheel, steering action -> moves chassis (because of the

offset, d)

• Meaning: for any chassis motion there exists some value for spin

speed 𝜑(t) and steering speed β(t) such that the constraints are met

• So, a robot with only castor wheels can move with any velocity in

the space of possible robot motions

• Example: five-castor wheel office chair• Can push it by hand in any direction

• Similarly, if 2-motors for any wheel any trajectory = possible

• Wheel kinematics almost complex, but do not impose any real constraints on the kinematics of a robot chassis

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Kinematics

Wheel kinematic constraints: 4-Swedish wheel

• No vertical axis of rotation, yet moves omnidirectionally

• by adding a DoF to the fixed standard wheel (rollers)

• Attached to wheel perimeter with antiparallel axes to main axis

• Angle = 𝜸 (between roller axes and the Wheel plane)

• Usually, 0 / 45 deg

• Since each axis can spin CW/CCW combine any vector

Along Axis-1 with any vector along axis-2 any direction

• Though Axes 1-2 not necessarily independent (except for 90-deg)

• Formulating constraint has some subtlety:• The instantaneous constraint is due to orientation of the small rollers.

• Rollers spin axis has zero component of velocity at the contact point (moving in that direction without spinning the main axis & without sliding is not possible)

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Kinematics

Wheel kinematic constraints: 4-Swedish wheel

• Motion constraint:• Looks identical to the rolling constraint for the fixed standard wheel

• except that formula is modified by adding 𝜸

• Therefore, effective direction along which the rolling constraint holds:

Along this zero component rather than along the wheel plane

• Orthogonal to this direction the motion is not constrained because of the free rotation of rollers

• Behavior of this constraint and thereby the Swedish wheel changes dramatically as 𝜸 varies

• Example: 𝜸 = 0

• rolling constraint = like fixed standard wheel

• No sliding constraint

• 𝜸 = 90

• No sliding constraint

• No benefit (in terms of lateral freedom of motion)

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Kinematics

Wheel kinematic constraints: 4-Swedish wheel

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Kinematics

Wheel kinematic constraints: 5-Spherical wheel

• Ball or spherical wheel• No direct constraints on motion

• No principal axis of rotation

• No appropriate rolling or sliding constraints

• Clearly omnidirectional as previous 2-types

• Describes the roll rate of the ball in the direction

of motion VA

• By definition the wheel rotation orthogonal to this

direction = 0:

Special case = moving along YR

• Equations for the spherical wheel are exactly the same as for the fixed standard wheel • However, the interpretation is different: The omnidirectional spherical wheel can have any arbitrary direction of

movement, where the motion direction given by β is a free variable

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Kinematics