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Robotics and Autonomous Systems

Lecture 5: Kinematics

Richard Williams

Department of Computer Science

University of Liverpool

• We’re still on the bottom right hand side.

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Kinematics Kinematics

• So far we have looked at different kinds of motion in a qualitative way.

• Two aspects to motion: • One way to program robots to move is trial and error.
• Locomotion
• Kinematics • A somewhat better way is to establish mathematically how the robot
should move, this is kinematics.
• Locomotion: What kinds of motion are possible?
• Rather kinematics is the business of figuring how a robot will move if
• Locomotion: What physical structures are there?
its motors work in a given way.
• Kinematics: Mathematical model of motion.
• Inverse-kinematics then tells us how to move the motors to get the
• Kinematics: Models make it possible to predict motion. robot to do what we want.
• We’ll look at a few tiny bits of the kinematics world.

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A formal model No kinematic model needed

• We will assume, as people usually do, that the robot’s location, or

pose is fixed in terms of three coordinates:

pxI , yI , θq

• Given that the robot needs to navigate to a location pxG , yG , θG q, it can

determine how x, y and θ need to change.
• BUT it can’t control these directly.
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Kinematic model Forward kinematics

• All a robot has access to are the speeds of its wheels:

ϕ91 , . . . ϕ9n • Together these determine the motion of the robot:

» fi
The steering angle of the steerable wheels: x9I
f pϕ91 , . . . ϕ9n , β1 , . . . βm , β1 , . . . βm q “
9 9 – y9I fl
β1 , . . . βm θ9

and the speed with which those steering angles are changing. • Forward kinematics

β91 , . . . β9m

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Reverse kinematics Reverse kinematics

• We can get what we want from the forward kinematic model:

• The forward kinematic model: ϕ91

» fi
— .. ffi
— . ffi
» fi
x9I
f pϕ91 , . . . ϕ9n , β1 , . . . βm , β91 , . . . β9m q “ – y9I fl — ϕ9n ffi
— ffi

θ9 — β1 ffi
— ffi
— .. ffi
— ffi
— . ffi “ f ´1 px9I , y9I , θq
9
is not what we want.
— βm ffi
— ffi
• We want to know how to set ϕ9i etc to get a given: — ffi
— β91 ffi
— .. ffi
— ffi
x9I , y9I , θ9 – . fl
β9m

• Reverse kinematics

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Three problems in kinematics Representing robot position

• The robot knows how it moves relative to its center of rotation.

• This is not the same as knowing how it moves relative to the world.

• Transformation between frames.

• Reversing the kinematic model.
Two systems of coordinates:
• Deriving robot motion from robot structure.
• Initial Frame: tXI , YI u

• Robot Frame: tXR , YR u

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Frame transformation Frame transformation

• In other words
• Robot position:
» fi » fi
x9 R x9 I
ξI “ rxI , yI , θI s T
– y9 R fl “ R pθq – y9 I fl
θ9 R θ9 I
• Mapping between frames: » fi » fi
cospθq sinpθq 0 x9 I
ξ9R “ R pθqξ9I “ – ´ sinpθq cospθq 0 fl – y9 I fl
“ R pθq x9I , y9I , θ9I
“ ‰T 0 0 1 θ9 I

meaning that:
where
x9 R “ x9 I cospθq ` y9 I sinpθq ` θ9 I .0
» fi
cospθq sinpθq 0
R pθq “ – ´ sinpθq cospθq 0 fl y9 R “ ´x9 I sinpθq ` y9 I cospθq ` θ9 I .0
0 0 1 θ9 R “ x9 I .0 ` y9 I .0 ` θ9 I .1

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Frame transformation Frame transformation

• In other words
» fi » fi
x9 R x9 I
– y9 R fl “ R pθq – y9 I fl • On other words, given how the robot moves in the world, we can
θ9 R θ9 I compute how the robot moves relative to its centre of rotation.
» fi » fi
cospθq sinpθq 0 x9 I • This is (part of) the forward kinematic model, but this isn’t what we
“ – ´ sinpθq cospθq 0 fl – y9 I fl want.
0 0 1 θ9 I • We want to be able to calculate how the robot moves in the world,
given how it moves relative to its centre of rotation.
meaning that:
• That is we want the reverse of this model.
x9 R “ x9 I cospθq ` y9 I sinpθq
y9 R “ ´x9 I sinpθq ` y9 I cospθq
θ9 R “ θ9 I

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Reverse kinematics Reverse kinematics

• To do this we compute:
• We want the reverse kinematic model:
» fi » fi
»
x9 I
fi
x9 R
» fi x9 I x9 R
– y9 I fl “ R pθq´1 – y9 R fl – y9 I fl “ R pθq´1 – y9 R fl
θ9 I θ9 R θ9 I θ9 R
» fi » fi
cospθq ´ sinpθq 0 x9 R
where R pθq´1 is the inverse of R pθq. “ – sinpθq cospθq 0 fl – y9 R fl
• Often R pθq´1 is hard to compute, but luckily for us in this case it isn’t. 0 0 1 θ9 R
• We have: »
cospθq ´ sinpθq 0
fi meaning that:
´1
R pθq “ – sinpθq cospθq 0 fl
x9 I “ x9 R cospθq ´ y9 R sinpθq ` θ9 R .0
0 0 1
y9 I “ x9 R sinpθq ` y9 R cospθq ` θ9 R .0
which we can use to establish x9I , y9I , θ9I θ9 I “ x9 R .0 ` y9 R .0 ` θ9 R .1

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Reverse kinematics Down to the structure of the robot

• To do this we compute:
» fi » fi
x9 I x9 R
– y9 I fl “ R pθq´1 – y9 R fl
θ9 I θ9 R
» fi » fi
cospθq ´ sinpθq 0 x9 R
“ – sinpθq cospθq 0 fl – y9 R fl
0 0 1 θ9 R

meaning that:

x9 I “ x9 R cospθq ´ y9 R sinpθq
• We can now identify the motion of the robot, in the global frame, if we
y9 I “ x9 R sinpθq ` y9 R cospθq know:
θ9 I “ θ9 R . x9 R , y9 R , θ9
but how do we tell what these are?
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Down to the structure of the robot Down to the structure of the robot

• We compute them from what we can measure, like the speed of the
wheels: • Some assumptions — constraints on the motion of the robot:
• Movement on a horizontal plane
• Point contact of the wheels; wheels not deformable
• Pure rolling, so 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)
• These won’t always be true, why?

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Differential drive Differential drive

• Consider differential drive.

• Wheels rotate at ϕ9 • Now, motion in the θ direction.
radians per second • Rotation due to right wheel is
• Each wheel contributes:
r ϕ9
r ϕ9 ωr “
2l
2
counterclockwise about the left
to motion of center of rotation. wheel.
• Motion in the x direction. • l is the distance between
• Total speed is the sum of two wheels.
contributions.

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Differential drive Differential drive

• Combining these components

we have:
• Rotation due to the left wheel is:
r ϕ9r r ϕ9l
» fi » fi
x9 R `
´r ϕ9 2 2
ωl “ – y9 R fl “ — 0
ffi
– fl
2l r ϕ9r r ϕ9l
θ9 R 2l ´ 2l
counterclockwise about the right
wheel. • And we can combine these with
R pθq´1 to find motion in the
global frame.

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Kinematic assumptions Kinematic assumptions

• Making sure the assumptions hold • Knowing what the assumptions are:
imposes constraints on robot imposes constraints on the applicability of the model.
• For example, ensuring a rigid chassis. • For example, ensuring wheels don’t slip.

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Why you keep track of assumptions More complex scenarios

Steered standard wheel Caster

• More parameters.
• Only fixed and steerable standard wheels impose constraints.

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Robot mobility Robot mobility

• The sliding constraint means that a standard wheel has no lateral

motion.
• A differential drive robot has just one line of zero motion.
• Zero motion line through the axis.

Instantaneous
center of rotation
• Thus its rotation is not constrained
• It can move in any circle it wants.
• Makes it very easy to move around.

• Has to move along a circle whose center is on the zero motion line.

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Robot mobility Robot mobility

• Formally we have the notion of a degree of mobility

• In general, the maneuverability of a robot depends on the number of
independent kinematic constraints. δm “ 3 ´ number of independent kinematic constraints
• Q: How can we formalise this idea?
• This number is also the number of independent fixed or steerable
• A: Degrees of mobility and maneuverability. standard wheels.
• The independence is important.

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Robot mobility Robot mobility

• Differential drive has two standard wheels, but they are on the same
axis. • A bicycle has two independent wheels, so two constraints.
• So not both independent.
• δm “ 1
• Number of constraints is 1.
• Can only alter x9 using wheel velocity.
• So δm “ 2 for a differential drive robot
• Can alter x9 and θ9 just through wheel velocity.

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Steerability and maneuverability Robot maneuverability

• Steering has an impact on how the robot moves.

• The degree of steerability δs is then the number of independent
steerable wheels.
• Differential drive has no steering wheels
• Note that a steerable standard wheel will both reduce the degree of • δs “ 0
mobility and increase the degree of steerability.
• δm “ 2, as before.
• The degree of maneuverability is:
• Thus, δM “ 2.
δM “ δm ` δs

• δM tells us how many degrees of freedom a robot can manipulate.

• Two robots with the same δM aren’t necessarily equivalent (see on).

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Robot maneuverability Common configurations

• A bicycle has one steering wheel

• δs “ 1
• δm “ 1, as before
• Thus, δM “ 2.

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Robot maneuverability Holonomy

• A bicycle, has a δM “ 2 yet can position itself anywhere in the plane.

• δM “ 2 is an indication of how easy it is for a robot to move around. • But a bicycle only has one DOF that it can control directly (x).
• Compare with the number of DOF in the environment. • Differential DOF
• DDOF is always equal to δm
• 3 for the environments we care about.
• A general inequality:
• Differential drive and bicycle both have δM “ 2, but you drive them
very differently. DDOF ď δM ď DOF

• A robot with DDOF “ DOF is called holonomic

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Holonomy

• We could compute forward and reverse kinematics for robot arms,

and use these to decide how to rotate motors to move the hand.

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 Summary

• This lecture looked a bit at kinematics

• The business of relating what robots do in the world to what their
motors need to be told to do.
• We did a little maths, but most of the discussion was qualitative.
• The Autonomous Robotics book goes more into the mathematical
detail of establishing kinematic constraints.
• Next time we’ll look at what support LeJOS gives for handling motion.

• Of course, this is how we get robot legs to do what we want also.

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