Day 7: Forward and Inverse Kinematics of Manipulator

Day 7: Forward and Inverse Kinematics

of Manipulator 2018. 1. 10. (Thu.)

Dai Owaki, Ph. D.,

Assist. Prof. Neuro-robotics Lab. (Hayashibe Lab.),Dept. Robotics, Graduate School of Engineering,

Tohoku University, Japan

1

Schedule for Robotics II in Python3

7. Forward and Inverse Kinematics (1/10 Thu.)

8. Jacobians (1/17 Thu.)

9. Dynamics (1/21 Mon.)

10.Linear Control (1/28 Mon.)

11.Nonlinear Control (1/31 Thu.)

12.Force and Impedance Control (TBA) (2/4 Mon.)

What is Kinematics? 6

l Kinematics is a branch of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies (group of objects) without considering the forces that caused the motion.

l Kinematics, as a field of study, is often referred to as the “geometry of motion” and is occasionally seen as a branch of mathematics.

From Wikipedia

Kinematics is not kinetics (dynmics), where we study how forces act on bodies!!!

Forward and Inverse Kinematics 7

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Forward Kinematics

Inverse Kinematics

“Non-redundant” manipulator End effector: 2 degrees of freedom (DoF) Joint angles: 2 degrees of freedom (DoF)

x, y

<latexit sha1_base64="DYuhCKfGyqGt8XqT0QvLRW1MkTk=">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</latexit><latexit sha1_base64="Ci9NU9YE7fPiI6EvILy3uoREXoU=">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</latexit><latexit sha1_base64="Ci9NU9YE7fPiI6EvILy3uoREXoU=">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</latexit><latexit sha1_base64="99+ntdQ/rNFAbs1KQCMs5PBFXL0=">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</latexit>

✓1, ✓2<latexit sha1_base64="nJykIyM3AH8mwfBPH7l+h8WTslU=">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</latexit><latexit sha1_base64="yRbWbo2RH8SUnhjMS/b8LRBu3Wo=">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</latexit><latexit sha1_base64="yRbWbo2RH8SUnhjMS/b8LRBu3Wo=">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</latexit><latexit sha1_base64="+mF1OTlOIsbtCKHUgSnrNofOWiw=">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</latexit>

Joint angles

Position of End effector


Forward Kinematics for 2-link 2-joint Manipulator 8

x







Non-redundant manipulator End effector: 2 degrees of freedom (DoF) Joint angles: 2 degrees of freedom (DoF)

(x = L1 cos ✓1 + L2 cos(✓1 + ✓2)

y = L1 sin ✓1 + L2 sin(✓1 + ✓2)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>


(x1 = L1 cos ✓1

y1 = L1 sin ✓1<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(x1, y1)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Position of the tip of link 1:

Position of the end effector:

(✓1, ✓2) ! (x, y)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

www.oscillex.org/robotics2day79

10ForwardKinematics_2Links.py Python script #1-1 $ python ForwardKinematics_2Links.py

Setting for parameters and variables

Definition of “Forward Kinematics” function !!! Please make the following code !!!

L1 : link 1 lengthL2 : link 2 length Th1 : link 1 angle Th2 : link 2 angle

math.cos() or np.cos()math.sin() or np.cos()

11Python script #1-2 ForwardKinematics_2Links.py $ python ForwardKinematics_2Links.py

Update function for th[0] (link 1 angle) by using slider1

Update function for th[1] (link 2 angle) by using slider2

12Python script #1-3 ForwardKinematics_2Links.py $ python ForwardKinematics_2Links.py

Setting and plotting for graph in figure

Setting for slider in figure

Update variable and plot the graph

Inverse Kinematics for 2-link 2-joint Manipulator 13

x









(x, y) ! (✓1, ✓2)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

O

A

P

�<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>


↵<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(✓1 = � � �

✓2 = ⇡ � ↵<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

OP = L =p

x2 + y2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

From “Cosine Theorem”,

L2= L2

1 + L22 � 2L1L2 cos↵

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

L22 = L2

1 + L2 � 2L1L cos�<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

cos� =

L21 + L2 � L2

2

2L1L<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

cos↵ =

L21 + L2

2 � L2

2L1L2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

tan � =y

x


Inverse Kinematics for 2-link 2-joint Manipulator 14

x










O

A

P




(✓1 = � � �


By using “inverse trigonometric functions”,

✓1 = arctan

y

x

� arccos

L

21 + L

2 � L

22

2L1L<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

✓2 = ⇡ � arccos

L21 + L2

2 � L2

2L1L2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

O

A

P

Multiple Solutions 15

x












(✓1 = � � �


or

l The gray lines indicate a second possible configuration in which the same end-effector position is achieved

l The number of solutions depends on the number of joints in the manipulator.

To select solution, we use some constrains, e.g.,

0 ✓2 ⇡<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

16InverseKinematics_2Links.py Python script #2-1 $ python InverseKinematics_2Links.py


Definition of “Inverse Kinematics” function !!! Please make the following code !!!

L1 : link 1 lengthL2 : link 2 length x: x-position of end effector y: y-position of end effector th1 : link 1 angle th2 : link 2 angle

math.pi or np.pimath.acos() or np.acos()math.atan2(y, x) or np.atan2(y, x)

“acos(x)” Function in Python: Technical Issue 17

The range of the argument “x” for acosfunction in python is limited from -1 to 1.

So, if we set the value out of the range, we found it make “error” as I will show it.

It would be not so serious issue, …

If you avoid this, please make some branch conditions as an right hand code.

Then,Output of acos function is also limited from 0 to pi radian.

So, it will automatically limit our multiple solutions of inverse kinematics for our 2-joint manipulator.

Forward Kinematics for 3-link 3-joint Manipulator 18

“Redundant” manipulator End effector: 2 degrees of freedom (DoF) Joint angles: 3 degrees of freedom (DoF)



x





✓3<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(✓1, ✓2, ✓3) ! (x, y)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(x = L1 cos ✓1 + L2 cos(✓1 + ✓2) + L3 cos(✓1 + ✓2 + ✓3)

y = L1 sin ✓1 + L2 sin(✓1 + ✓2) + L3 sin(✓1 + ✓2 + ✓3)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>



L3<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

��

ForwardKinematics_3Links.py Python script #3-1 $ python ForwardKinematics_3Links.py


Definition of “Forward Kinematics” function !!! Please make the following code !!!

L1 : link 1 lengthL2 : link 2 length L3 : link3 lengthTh1 : link 1 angle Th2 : link 2 angle Th3 : link 3 angle


Forward Kinematics for 3-link 3-joint Manipulator in 3D 20


x







(✓yaw, ✓1, ✓2) ! (x, y, z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

✓yaw<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

8><

>:

x = {L1 cos ✓1 + L2 cos(✓1 + ✓2)} cos ✓yawy = {L1 cos ✓1 + L2 cos(✓1 + ✓2)} sin ✓yawz = L1 sin ✓1 + L2 sin(✓1 + ✓2)


��

ForwardKinematics_2Links_3D.py Python script #4-1 $ python ForwardKinematics_2Links_3D.py


Definition of “Forward Kinematics” function

L1 : link 1 lengthL2 : link 2 length Th1 : link 1 angle yaw Th2 : link 1 angle 1Th3 : link 2 angle 2


Kinematics for “Redundant” Manipulators 22

x








x







✓4<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>? ?

? ?

Forward Kinematics Inverse Kinematics

Geometrical solution is UNIQUELY computable: Joint angles to End effector position

Geometrical solution is NOT uniquely computable: End effector position to Joint angles

End effector: 2 degrees of freedom (DoF) Joint angles: 4 degrees of freedom (DoF)

Report #1: Deadline 1/17 Thu. 23

Kinematics exercise:

1. Make a python script for forward kinematics of “more than 4-joint” manipulator.

2. Calculate forward and inverse kinematics for two examples, which I will explain in the next slide, and make python scripts for them (you can download templates for these scripts).

Please send it me by e-mail ([email protected]), or put printed one in a report box @ A15 503 (5F): If you need a few feedback, please (correctly) write your email address!!!

Exercise 1 24

(l, ✓) $ (x, y)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(x =

y =<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(l =

✓ =<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Forward kinematics Inverse kinematics

x



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ForwardKinematics_2Links_Exc1.py InverseKinematics_2Links_Exc1.py

Exercise 2 25

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✓<latexit sha1_base64="MmLgCmxWZ8ev+3iessedmwxp+nE=">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</latexit><latexit sha1_base64="pa+ni+DsQbOJq8Gbdnbf7eJDYSM=">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</latexit><latexit sha1_base64="pa+ni+DsQbOJq8Gbdnbf7eJDYSM=">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</latexit><latexit sha1_base64="5CsmxDbDInG8yPnWWf31N00Mr7Y=">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</latexit>


L<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(x =

y =<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(l, ✓) $ (x, y)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(l =

✓ =<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Forward kinematics Inverse kinematics

ForwardKinematics_2Links_Exc2.py InverseKinematics_2Links_Exc2.py

End.

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Documents

Day 7: Forward and Inverse Kinematics of Manipulator