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An Introduction to

Robot Kinematics

© Howie Choset, 2019

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We are interested in two kinematics topics

Forward Kinematics (angles to position)What you are given: The length of each link

The angle of each joint

What you can find: The position of any point

(i.e. it’s (x, y, z) coordinates

Inverse Kinematics (position to angles)

What you are given: The length of each link

The position of some point on the robot

What you can find: The angles of each joint needed to obtain

that position

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2

1

(x , y)

l2

l1

Inverse Kinematics of a Two Link Manipulator

Given: l1, l2 , x , y

Find: 1, 2

Redundancy:

A unique solution to this problem

does not exist. Notice, that using the

“givens” two solutions are possible.

Sometimes no solution is possible.

(x , y)

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The Geometric Solution

Using the Law of Cosines:

21

2

2

2

1

22

21

2

2

2

1

22

21

2

2

2

1

22

222

2arccosθ

2)cos(θ

)cos(θ)θ180cos(

)θ180cos(2)(

cos2

ll

llyx

ll

llyx

llllyx

Cabbac

2

2

22

2

Using the Law of Sines:

x

y2arctanα

θαθ

yx

)sin(θ

yx

)θsin(180θsin

sinsin

11

22

2

22

2

2

1

l

c

C

b

B

l1

l2 q2

q1

(x , y)

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The Geometric Solution

Using the Law of Cosines:

21

2

2

2

1

22

21

2

2

2

1

22

21

2

2

2

1

22

222

2arccosθ

2)cos(θ

)cos(θ)θ180cos(

)θ180cos(2)(

cos2

ll

llyx

ll

llyx

llllyx

Cabbac

2

2

22

2

Using the Law of Sines:

x

y2arctanα

θαθ

yx

)sin(θ

yx

)θsin(180θsin

sinsin

11

22

2

22

2

2

1

l

c

C

b

B

22

221

yx

)sin(θarcsin,2arctanθ

lxy

Redundant since q2 could be in the

first or fourth quadrant.

Redundancy caused since q2 has two possible

values

l1

l2 q2

q1

(x , y)

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21

2

2

2

1

22

2

221

2

2

2

1

21121121

2

2

2

1

21121

2

21

2

2

2

1

2

121121

2

21

2

2

2

1

2

1

2222

2

yxarccosθ

c2

)(sins)(cc2

)(sins2)(sins)(cc2)(cc

yx)2((1)

ll

ll

llll

llll

llllllll

The Algebraic Solution

l1

l2 q2

q1

(x , y)

21211

21211

1221

11

sinsy(2)

ccx(1)

)θcos(θc

cosθc

ll

ll

Only Unknown

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

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

:

abbaba

bababa

Note

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

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

:

abbaba

bababa

Note

)c(s)s(c

cscss

sinsy

)()c(c

ccc

ccx

2211221

12221211

21211

2212211

21221211

21211

lll

lll

ll

slsll

sslll

ll

We know what q2 is from the previous

slide. We need to solve for q1 . Now

we have two equations and two

unknowns (sin q1 and cos q1 )

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

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

:

abbaba

bababa

Note

)c(s)s(c

cscss

sinsy

)()c(c

ccc

ccx

2211221

12221211

21211

2212211

21221211

21211

lll

lll

ll

slsll

sslll

ll

We know what q2 is from the previous

slide. We need to solve for q1 . Now

we have two equations and two

unknowns (sin q1 and cos q1 )

1

1

22122

22221

s

c

cs

sc

lll

lll

y

x

),(2arctan 111 csq

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Three-link Manipulator IK

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RPR Arm

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RPR Arm

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RPR

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RPR

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Workspace of Two-Link

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Workspace of Two-Link

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PRR Arm

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PRR Arm

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PRR Arm

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PRR Arm

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PRR Arm

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