robot dynamics – newton- euler recursive approach
DESCRIPTION
Robot Dynamics – Newton- Euler Recursive Approach. ME 4135 Robotics & Controls R. Lindeke, Ph. D. Physical Basis:. This method is jointly based on: Newton’s 2 nd Law of Motion Equation: and considering a ‘rigid’ link Euler’s Angular Force/ Moment Equation: . - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/1.jpg)
Robot Dynamics – Newton- Euler Recursive Approach
ME 4135 Robotics & ControlsR. Lindeke, Ph. D.
![Page 2: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/2.jpg)
Physical Basis:
This method is jointly based on:– Newton’s 2nd Law of Motion
Equation:
and considering a ‘rigid’ link
– Euler’s Angular Force/ Moment Equation:
i CF m
i imoment CM i i CM iN I I
![Page 3: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/3.jpg)
Again we will Find A “Torque” Model
Each Link Individually We will move from Base to End to find Velocities
and Accelerations We will move from End to Base to compute force
(f) and Moments (n) Finally we will find that the Torque is:
1 1(1 )i i iqT T
n z f z bi i i i i i
Gravity is explicitly included in the model by considering acc0 = g where g is (0, -g0, 0) or (0, 0, -g0)
i is the joint type parameter (is 1 if
revolute; 0 if prismatic) like in
Jacobian!
![Page 4: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/4.jpg)
Lets Look at a Link “Model”
X0
Y0
Z0
Zk-1
Xk-1Yk-1
ZK
XK
YK
dK
dK-1
sK
![Page 5: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/5.jpg)
We will Build Velocity Equations
Consider that i is the joint type parameter (is 1 if revolute; 0 if prismatic)
Angular velocity of a Frame k relative to the Base:
NOTE: if joint k is prismatic, the angular velocity of frame k is the same as angular velocity of frame k-1!
1 1k k k k kq z
![Page 6: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/6.jpg)
Angular Acceleration of a “Frame”
Taking the Time Derivative of the angular velocity model of Frame k:
1 1 1 1k k k k k k k kq z q z
Same as (dw/dt) the angular acceleration in
dynamics
![Page 7: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/7.jpg)
Linear Velocity of Frame k:
Defining sk = dk – dk-1 as a link vector, Then the linear velocity of link K is:
Leading to a Linear Acceleration Model of:
1 11k k k k k k kv v s q z
1
1 11 2k k k k k k k
k k k k k k
s s
q z q z
Normal component of acceleration (centrifugal
acceleration)
![Page 8: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/8.jpg)
This completes the Forward Newton-Euler Equations:
To evaluate Link velocities & accelerations, start with the BASE (Frame0)
Its Set V & A set (for a fixed or inertial base) is:
As advertised, setting base linear acceleration propagates gravitational effects throughout the arm as we advance toward the end!
0
0
0
0
000vg
![Page 9: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/9.jpg)
Now we define the Backward (Force/Moment) Equations
Work Recursively from the End We define a term rk which is the vector from
the end of a link to its center of mass:
k k k
k
r c dc
is location of center of mass of Link k
![Page 10: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/10.jpg)
Defining f and n Models
1k k k k k k k k kf f m r r
1 1k k k k k k k k k
k k k
n n s r f r f D
D
Inertial Tensor of Link k – in base space
The term in the brackets represents the linear acceleration of the center of mass of Link k
![Page 11: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/11.jpg)
Combine them into Torque Models:
1 11T Tk k k k k k k k kn z f z b q
NOTE: For a robot moving freely in its workspace without
carrying a payload, ftool = 0
We will begin our recursion by setting fn+1 = -ftool and nn+1= -ntool
Force and moment on the tool
![Page 12: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/12.jpg)
The overall N-E Algorithm:
Step 1: set T00 = I; fn+1 = -ftool; nn+1 = -ntool; v0 = 0;
vdot0 = -g; 0 = 0; dot
0 = 0 Step 2: Compute –
Zk-1’s Angular Velocity & Angular Acceleration of Link k Compute sk
Compute Linear velocity and Linear acceleration of Link k Step 3: set k = k+1, if k<=n back to step 2 else set k = n
and continue
![Page 13: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/13.jpg)
The N-E Algorithm cont.:
Step 4: Compute – rk (related to center of mass of Link k)fk (force on link k)
Nk (moment on link k)tk (torque on link k)
Step 5: Set k = k-1. If k>=1 go to step 4
1 10 0
T
k kD DR R
![Page 14: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/14.jpg)
So, Lets Try one:
Keeping it Extremely Simple
This 1-axis ‘robot’ is called an Inverted Pendulum
It rotates about z0 “in the plane” x0-y0 X0
Z0
Y0m1
Y1
X1
Z1
![Page 15: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/15.jpg)
Writing some info about the device:1
1
21 1
1
1 1 1 1
1 1 1 110 1
2001
0 0 00 1 0
120 0 1
00
0 0 1 00 0 0 1
l
c
m lD
C S l CS C l S
T A
“Link” is a thin cylindrical rod
![Page 16: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/16.jpg)
Continuing and computing:
1 1 10 0
11 1 1 1
1 1 1 1
1 1
1 1
0 21 0 0 00 00 1 0 0
0 0 1 0 00 0 1 00 0 0 1 1
2
20
c H T c
lC S l CS C l S
l C
l S
![Page 17: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/17.jpg)
Inertial Tensor computation:
1 11 0 1 0
1 1 1 121 1
1 1 1 1
21 1 12
21 11 1 1
0 0 0 0 00 0 1 0 0
120 0 1 0 0 1 0 0 1
00
120 0 1
TD R D R
C S C Sm lS C S C
S S Cm l S C C
![Page 18: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/18.jpg)
Let’s Do it (Angular Velocity & Accel.)!
1 1 1
1 1 1
1 1
1 111 0
1 1
1 1
0 00 0 0
1 1
0 00 0 0
1 1
1 0 0 0( ) 0 1 0 0
00 0 1 0
1
0
o
q q
q q
l Cl S
s H O O
l Cl S
Starting: Base (i=0)Ang. vel = Ang. acc = Lin. vel = 0
Lin. Acc = -g which is (0, -g0, 0)T
1 = 1
![Page 19: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/19.jpg)
Linear Velocity:
1 0 1 1 1 1 0
1
1 1 1 1
1
1 1 1
1
0 00 0 1 1 0
1 0 1
0
v v s q z
Cq l S q
Sl q C
![Page 20: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/20.jpg)
Linear Acceleration: 1 1 1 1 1 1
1 1
1 1 1 1 1 1 1
1 12
1 1 1 1 1 1
1 12
1 1 1 1 1 1
00
0 1 0
0 0
0 0
g s s
S Sg l q C q l q C
S Cg l q C l q S
S Cg l q C l q S
Note: g = (0, -g0, 0)T
![Page 21: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/21.jpg)
Thus Forward Activities are done!
Compute r1 to begin Backward Formations:
1 1 1
1 1
1 1
1 11 1
1
11
2
200
20
r c Ol C
l Cl S l S
Cl S
![Page 22: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/22.jpg)
Finding f1
Consider: ftool = 0
1 1 1 1 1 1 1
1 11 1 1 1
1 1 1 1 1
11 1
1 1 1
{ }
0 0 00 0 0
2 21 0 1 1 0
20
f m r r
C Cl q l qm S q S
Sl qm C
12
1 11
1 121 1 1 1
1 1 1 1
1 12 1 1
1 1 1 1 1 1 1
00
21 0
2 20 0
20 0
Sl q C
S Cl q l qm C S
S C Sl qm g l q C l q S
1 12
1 11 120 0
Cl qC S
![Page 23: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/23.jpg)
Collapsing the terms
1 121 1 1 1
1 1 1 1
21 1 1 1 1
21 1 1 1 1
1 1 0
2 20 0
2
20
S Cl q l qf m g C S
l q S q C
l q C q Sf m g
simplifying:
![Page 24: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/24.jpg)
Computing n1:
1 1 1 1 1 1 1 1 1
1 1 2 2 21 1 1 1 1 1 1
1 1 1 1
21 1 1 1 1
211 1 1 1 11
1 1 0
0 0 00 0 0
2 12 120 0 1 1 1
2
2 20
n s r f D D
C Cl m l q m l ql S S f
l q S q C
Cl q C q Sl S m g
21 1 1
2 2 21 1 1 1 1 1 1 1 1 1 1 1 11 1
1 0 1
00
1210
0 00 02 2 2 121 1
m l q
l q C q S l q S q C m l ql m C g S
This X-product goes to Zero!
The Link Force Vector
![Page 25: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/25.jpg)
Simplifying:
21 0 1 1 11 1 1
00
3 21
m g m l Cn l q
![Page 26: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/26.jpg)
Writing our Torque Model
1 1 1 0 1 1 0 1 1
21 0 1 1 11 1 1 1
21 1 0 1 1 1
1 1 1 1
1
0 01 0 0 0 1 1 0
3 21 1
3 2
T T
T
n z f z b q
m g m l Cl q f b q
m l g m l Cq b q
‘Dot’ (scalar) Products
![Page 27: Robot Dynamics – Newton- Euler Recursive Approach](https://reader033.vdocument.in/reader033/viewer/2022061423/5681619e550346895dd15534/html5/thumbnails/27.jpg)
Homework Assignment (mostly for practice!):
• Compute L-E solution for “Inverted Pendulum & Compare torque model to N-E solution
• Compute N-E solution for 2 link articulator (of slide set: Dynamics, part 2) and compare to our L-E torque model solution computed there
• Consider Our 4 axis SCARA robot – if the links can be simplified to thin cylinders, develop a generalized torque model for the device.