Joint Trajectory for Robot Circular path

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Somasundar Avantsa

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Sep 21, 2017, 11:57:19 PM9/21/17
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Hi,

While trying 6DOF Kuka KR5 Arc Robot's Joint trajectories calculation in Matlab I am facing a problem.

For a given Circular path (in XY Plane) to be traced out by End Effector, and Initial Pose of Robot, I am finding Jacobian Matrix at each successive point on Circular path and joint angles using the code based on this logic.

deltaTheta = InverseOfJacobian*deltaX; where deltaX has delta increments along X, Y, Z and  Roll,Pitch, Yaw ( Fixed Orientation of End Effector for entire path)

But while inputting this Joint trajectories to Robot Simulation, Circular path is not achieved by End Effector.

Can any one please help?

Matlab Code:

% 6DOF Kuka KR5 Arc Robot D_H Notation and Robot COnfiguration
close all;
clear all;

l=Link([0,.4,.18,(pi)/2,0])
l(2)=Link([0,.135,.6,(pi),0])
l(3)=Link([0,.135,.12,-(pi)/2,0])
l(4)=Link([0,.62,0,(pi)/2,0])
l(5)=Link([0,0,0,-(pi)/2,0])
l(6)=Link([0,0.115,0,0,0])
R=SerialLink(l)


%for circle
A = rand(2);
m = 200;
angles = linspace(0,2*pi,m);
f = @(theta) [cos(theta),sin(theta)];
r = arrayfun(@(i)f(angles(i)),1:m,'UniformOutput',false);
pt = reshape(cell2mat(r),[2 m]);
x= pt(1,:)
y= pt(2,:)


line(pt(1,:),pt(2,:));
hold on
plot(pt(1,:),pt(2,:),'ro')

% Initial joint angles
th = [0 0 0 0 (30*pi)/180 0].'


%Robot Pose [x y z a b c]

%p1 = [3.1416 0.5236 0 .8425 0 -.3196].'
p = [ .8425 0 -.3196 3.1416 0.5236 0].'


dth= [0 0 0 0 0 0].'


% to populate joint angles at each interval of ee path by 
for i=1:200
    j=R.jacob0(p, 'rpy')
       
    ji=inv(j)
    %dp1=[0 0 0 .0025 yy(i+1)-yy(i) 0].'
    dp=[x(i+1)-x(i) y(i+1)-y(i) 0 0 0 0].'
    y(i+1)
    y(i)    
    dth=ji*dp
    th=th+dth
    p=p+dp
    th1=th.'
    th1 = th1*180/pi
    mycell=sprintf('A%s', num2str(i+1))
    xlswrite('Circle1.xlsx',th1,1,mycell)
end


with regards,
Somasundar
circle_expected.png
path_achieved.png

Peter Corke

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Nov 11, 2017, 8:02:08 PM11/11/17
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I can sort of see what you're trying to do, resolved rate motion control around a circle.  What's the difference between p and th?  This bit I don't get.

Why not use inverse kinematics to do this instead?

Somasundar Avantsa

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Nov 13, 2017, 4:00:53 AM11/13/17
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Sir,

th is the vector of intial jint angles - [ theta1, theta2 upto theta6]
p is the End Effector pose [ position orientation] = [x y z roll pitch yaw]

I could resolve this problem by help from the forum. 
I replaced p with th in the following line and got the result
j=R.jacob0(p, 'rpy')

however, How can I do this with Inverse kinematics because there are multiple vectors of joint angles for a single end effector pose that results from Inverse kinematics.
how to choose the right one at each intermediate point of the circular trace of End Effector?

Please help.

regards,
Soma sundar A

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Peter Corke

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Nov 13, 2017, 4:46:34 AM11/13/17
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On Monday, November 13, 2017 at 7:00:53 PM UTC+10, Somasundar Avantsa wrote:
Sir,

th is the vector of intial jint angles - [ theta1, theta2 upto theta6]
p is the End Effector pose [ position orientation] = [x y z roll pitch yaw]

I could resolve this problem by help from the forum. 
I replaced p with th in the following line and got the result
j=R.jacob0(p, 'rpy')

that makes more sense, the Jacobian is computed from joint angles not pose.
 

however, How can I do this with Inverse kinematics because there are multiple vectors of joint angles for a single end effector pose that results from Inverse kinematics.
how to choose the right one at each intermediate point of the circular trace of End Effector?

Compute a vector of end-effector poses and use the ikine() method, it will return a set of joint angles.  Yes there are multiple solutions but ikine() will return the one closest to the initial estimate you give. For a vector of poses, for each pose it uses the set of joint angles from the previous pose as the initial estimate.

Somasundar Avantsa

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Aug 7, 2018, 3:45:06 AM8/7/18
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Dear Sir,

I got the Jacobian Inverse approach right with your guidance and completed path tracing by KUKA KR5 Arc Model for some curves.

Now I am trying the same by Inverse Kinematics. However in IK, I am getting curve not as expected. Please find code below and images attached

% D-H parameters using 
%startup_rvc
%L = Link ( [Th   d   a   alph] 'Link'  );
 L(1) = Link  ( [ 0 .4 .18   pi/2] );
 L(2) = Link  ( [ 0 .135  .6  pi   ] );
 L(3) = Link  ( [ 0 .135  .12  -pi/2 ] );
 L(4) = Link  ( [ 0 .62  0   pi/2] );
 L(5) = Link  ( [ 0 0  0  -pi/2   ] );
 L(6) = Link  ( [ 0 .115  0   0 ] );

  Rob = SerialLink (L);
 A= cat(3, [1 0 0 1.296; 0 -1 0 0; 0 0 -1 .18361; 0 0 0 1], [ 1 0 0 1.896; 0 -1 0 0; 0 0 -1 .18361; 0 0 0 1], [ 1 0 0 1.896; 0 -1 0 0; 0 0 -1 .48361; 0 0 0 1], [ 1 0 0 1.296; 0 -1 0 0; 0 0 -1 .48361; 0 0 0 1]);
 %[ 1 0 0 1.296; 0 -1 0 0; 0 0 -1 .18361; 0 0 0 1]);
   q1=Rob.ikine(A);
 Rob.plot(q1,'trail','r');


Kindly suggest.

Regards,
Somasundar 


fig_expected.png
fig_output.png

Somasundar Avantsa

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Aug 8, 2018, 12:23:08 AM8/8/18
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Sir,
 
I followed as per your earlier disucssion in this thread and wrote IK code.  But it does not seem to go to the next nearest joint configuration in IK algorithm and the out put path is different than that expected.  Kindly help me to resolve this problem.

with regards,
Somasundar A

Peter Corke

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Aug 9, 2018, 9:51:44 PM8/9/18
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Erik van Oene

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Aug 10, 2018, 5:32:21 PM8/10/18
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Your second pose is not reachable.

Erik

Op vrijdag 10 augustus 2018 03:51:44 UTC+2 schreef Peter Corke:
firstPose.PNG
secondPose.PNG

Somasundar Avantsa

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Aug 16, 2018, 1:52:34 AM8/16/18
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Dear Sir,

In place of ikine, I used ikcon and some errors were prevented. However, When I try to plot a Rectangle in Matlab using ikcon in XY Plane (rectangle in 2-D), It is plotting some rhombus like figure that too in 3-D. I verified my code and it seems alright. Not able to figure out where the problem might be. I am using toolbox 9.8

CODE:
% D-H parameters using 
%startup_rvc
%L = Link ( [Th   d   a   alph] 'Link'  );
 L(1) = Link  ( [ 0 .4 .18   pi/2] );
 L(2) = Link  ( [ 0 .135  .6  pi   ] );
 L(3) = Link  ( [ 0 .135  .12  -pi/2 ] );
 L(4) = Link  ( [ 0 .62  0   pi/2] );
 L(5) = Link  ( [ 0 0  0  -pi/2   ] );
 L(6) = Link  ( [ 0 .115  0   0 ] );

  Rob = SerialLink (L);
 A= cat(3, [1 0 0 1.2; 0 -1 0  0.5; 0 0 -1 .2; 0 0 0 1], [ 1 0 0 2.7; 0 -1 0 0.5; 0 0 -1 .2; 0 0 0 1],[ 1 0 0 2.7; 0 -1 0 1.3; 0 0 -1 .2; 0 0 0 1], [ 1 0 0 1.2; 0 -1 0 1.3; 0 0 -1 .2; 0 0 0 1], [ 1 0 0 1.2; 0 -1 0 0.5; 0 0 -1 .2; 0 0 0 1]);
 %);
 %t=0:.04:2;
   q1=Rob.ikcon(A);
 Rob.plot(q1,'trail','r');

Please find output screen shots

best regards,
Somasundar A








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op_fig1.png
op_fig2.png

Erik van Oene

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Aug 18, 2018, 10:25:52 AM8/18/18
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Hi,

2 questions:

1)  Why would you use ikcon?
2)  Why would pose 2, which is even further in X direction as before (2.7 instead of 1.896) , now be reachable if you robot parameters didn't change?




Op donderdag 16 augustus 2018 07:52:34 UTC+2 schreef Somasundar Avantsa:
Dear Sir,

In place of ikine, I used ikcon and some errors were prevented. However, When I try to plot a Rectangle in Matlab using ikcon in XY Plane (rectangle in 2-D), It is plotting some rhombus like figure that too in 3-D. I verified my code and it seems alright. Not able to figure out where the problem might be. I am using toolbox 9.8

CODE:
% D-H parameters using 
%startup_rvc
%L = Link ( [Th   d   a   alph] 'Link'  );
 L(1) = Link  ( [ 0 .4 .18   pi/2] );
 L(2) = Link  ( [ 0 .135  .6  pi   ] );
 L(3) = Link  ( [ 0 .135  .12  -pi/2 ] );
 L(4) = Link  ( [ 0 .62  0   pi/2] );
 L(5) = Link  ( [ 0 0  0  -pi/2   ] );
 L(6) = Link  ( [ 0 .115  0   0 ] );

  Rob = SerialLink (L);
 A= cat(3, [1 0 0 1.2; 0 -1 0  0.5; 0 0 -1 .2; 0 0 0 1], [ 1 0 0 2.7; 0 -1 0 0.5; 0 0 -1 .2; 0 0 0 1],[ 1 0 0 2.7; 0 -1 0 1.3; 0 0 -1 .2; 0 0 0 1], [ 1 0 0 1.2; 0 -1 0 1.3; 0 0 -1 .2; 0 0 0 1], [ 1 0 0 1.2; 0 -1 0 0.5; 0 0 -1 .2; 0 0 0 1]);
 %);
 %t=0:.04:2;
   q1=Rob.ikcon(A);
 Rob.plot(q1,'trail','r');

Please find output screen shots

best regards,
Somasundar A








On Sa
Your second pose is not reachable.

Erik

Op vrijdag 10 augustus 2018 03:51:44 UTC+2 schreef Peter Corke:
This posting might be relevant to your problem

https://groups.google.com/forum/#!msg/robotics-tool-box/VMIbW6psMPQ/5ZRvLoGtBAAJ;context-place=topic/robotics-tool-box/7bNUFrP_-8A

On Wednesday, 8 August 2018 00:23:08 UTC-4, Somasundar Avantsa wrote:
Sir,
 
I followed as per your earlier disucssion in this thread and wrote IK code.  But it does not seem to go to the next nearest joint configuration in IK algorithm and the out put path is different than that expected.  Kindly help me to resolve this problem.

with regards,
Somasundar A

Peter Corke

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Aug 19, 2018, 10:36:49 AM8/19/18
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Using RTB10.2 I see that the forward kinematics doesn't give the original poses

>> Rob.fkine(q1)
 

ans(1) = 
    1.0000   -0.0001    0.0001       1.2
   -0.0001   -1.0000   -0.0001       0.5
    0.0001    0.0001   -1.0000       0.2
         0         0         0         1

ans(2) = 
    0.9922    0.0014    0.1250     1.402
   -0.0015   -0.9997    0.0231    0.2596
    0.1250   -0.0231   -0.9919    0.2444
         0         0         0         1

ans(3) = 
    0.9908    0.0046    0.1354     1.287
   -0.0043   -0.9979    0.0652    0.6196
    0.1354   -0.0652   -0.9886    0.2483
         0         0         0         1

ans(4) = 
    0.9997    0.0003    0.0233    0.9589
   -0.0003   -0.9997    0.0253     1.039
    0.0233   -0.0253   -0.9994    0.2192
         0         0         0         1

ans(5) = 
    1.0000   -0.0001    0.0001       1.2
   -0.0001   -1.0000   -0.0000       0.5
    0.0001    0.0000   -1.0000       0.2
         0         0         0         1

as Erik suggest perhaps these points aren't reachable.

If you look at the solution error

>> [q1,err]=Rob.ikcon(A);
>> err

err =

    0.0000
    2.9895
    4.2166
    0.2171
    0.0000

it is again clear that there is a problem with reachability of points 2,3,4

Somasundar Avantsa

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Aug 27, 2018, 1:04:54 AM8/27/18
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Dear Sir,

I got corrected path by changing robot pose to within work volume as you suggested
Is there a tool/method to simply check whether a pose is singular or not?

thank you
Somasundar A

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Peter Corke

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Aug 27, 2018, 1:17:37 PM8/27/18
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A point being out of reach is sometimes referred to as a singularity, the only way to test for that is failure of IK.

A kinematic singularity happens when the determinant of the Jacobian matrix becomes zero.

peter
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Somasundar Avantsa

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Aug 28, 2018, 1:16:57 AM8/28/18
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Thank you Sir

regards,
Somasundar AV

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Somasundar Avantsa

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Sep 3, 2018, 6:00:59 AM9/3/18
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Dear Sir,

When I trace a Circular Arc End Effector path, by using ikcon,  
I am getting the issue detailed in post at URL  which you have shared earlier. However the End Effector's Trace of Poses is perfect. 
But the Joint trajectories are complex.
https://groups.google.com/forum/#!msg/robotics-tool-box/VMIbW6psMPQ/5ZRvLoGtBAAJ;context-place=topic/robotics-tool-box/7bNUFrP_-8A  
I am attaching image of the Robot Joint trajectory. Is it possible to get EE path traced by a Series of Simple Joint trajectories?  
Kindly help me to use appropriate method.

best regards,
Somasundar AV


Initial Pose.png
Final pose.png
Initial n Final Joint Angles.png

Somasundar Avantsa

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Sep 3, 2018, 6:21:06 AM9/3/18
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Dear Sir,

What I found for alternative IK Solutions for Initial Pose are attached. Out of the 8 alternatives 5 and 6 are simple. I guess the the EE Circular path 
has no singular points when I start my Joint trajectory from 5th or 6th Solutions. 

Kindly guide me to select this trajectory for tracing the path

best regards,
Somasundar AV
5th Configuration.png
Alternative IK Solutions for Initial Pose.png

Erik van Oene

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Sep 6, 2018, 4:00:51 AM9/6/18
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Hi,

What you could do is calculate the manipulability of the robot over whole path. The manipulability says something about how close the robot is to a singular point. When you know this over the whole path the 'optimal' path can be selected (optimal in the sense of avoiding singularity). Off cause you can extend this by avoiding joint limits, etcetera.

Erik


Op maandag 3 september 2018 12:21:06 UTC+2 schreef Somasundar Avantsa:
Dear Sir,

What I found for alternative IK Solutions for Initial Pose are attached. Out of the 8 alternatives 5 and 6 are simple. I guess the the EE Circular path 
has no singular points when I start my Joint trajectory from 5th or 6th Solutions. 

Kindly guide me to select this trajectory for tracing the path

best regards,
Somasundar AV

On Mon, Sep 3, 2018 at 3:30 PM Somasundar Avantsa <> wrote:
Dear Sir,

When I trace a Circular Arc End Effector path, by using ikcon,  
I am getting the issue detailed in post at URL  which you have shared earlier. However the End Effector's Trace of Poses is perfect. 
But the Joint trajectories are complex.
https://groups.google.com/forum/#!msg/robotics-tool-box/VMIbW6psMPQ/5ZRvLoGtBAAJ;context-place=topic/robotics-tool-box/7bNUFrP_-8A  
I am attaching image of the Robot Joint trajectory. Is it possible to get EE path traced by a Series of Simple Joint trajectories?  
Kindly help me to use appropriate method.

best regards,
Somasundar AV



On Tue, Aug 28, 2018 at 10:46 AM Somasundar Avantsa <> wrote:
Thank you Sir

regards,
Somasundar AV

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Somasundar Avantsa

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Sep 7, 2018, 4:37:03 AM9/7/18
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Dear Sir,
I don't know if you have read my earlier email as follows.

When I trace a Circular Arc End Effector path, by using ikcon,  
I am getting the issue detailed in post at URL  which you have shared earlier. However the End Effector's Trace of Poses is perfect. 
But the Joint trajectories are complex.
https://groups.google.com/forum/#!msg/robotics-tool-box/VMIbW6psMPQ/5ZRvLoGtBAAJ;context-place=topic/robotics-tool-box/7bNUFrP_-8A  
I am attaching image of the Robot Joint trajectory. Is it possible to get EE path traced by a Series of Simple Joint trajectories?  
Kindly help me to use appropriate method.

What I found for alternative IK Solutions for Initial Pose are attached. Out of the 8 alternatives 5 and 6 are simple. I guess the the EE Circular path 
has no singular points when I start my Joint trajectory from 5th or 6th Solutions. 

Kindly guide me to select this trajectory for tracing the path

Here as you suggested sir, I found Manipulability for Joint Configurations starting from alternative IK Solution 1(given in attachment) and I got them as 

Columns 1 through 12

    0.3719    0.3754    0.3784    0.3812    0.3835    0.3856    0.3874    0.3889    0.3902    0.3912    0.3920    0.3925

  Columns 13 through 24

    0.3929    0.3931    0.3931    0.3930    0.3927    0.3924    0.3920    0.3914    0.3908    0.3901    0.3894    0.3886

  Columns 25 through 36

    0.3878    0.3870    0.3862    0.3854    0.3845    0.3840    0.3831    0.3825    0.3820    0.3813    0.3807    0.3803

  Columns 37 through 48

    0.3798    0.3795    0.3793    0.3790    0.3789    0.3788    0.3788    0.3789    0.3792    0.3795    0.3796    0.3801

  Columns 49 through 60

    0.3806    0.3809    0.3813    0.3819    0.3824    0.3830    0.3837    0.3846    0.3851    0.3860    0.3867    0.3874

  Columns 61 through 72

    0.3880    0.3885    0.3890    0.3895    0.3898    0.3900    0.3901    0.3901    0.3899    0.3896    0.3891    0.3884

  Columns 73 through 84

    0.3874    0.3863    0.3848    0.3830    0.3810    0.3786    0.3760    0.3729    0.3695    0.3657    0.3612    0.3565

  Columns 85 through 96

    0.3512    0.3451    0.3386    0.3315    0.3236    0.3149    0.3052    0.2946    0.2828    0.2697    0.2551    0.2385

  Columns 97 through 100

    0.2182    0.1947    0.1602    0.1602




best regards,
Somasundar AV


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Initial Pose.png
Final pose.png
Initial n Final Joint Angles.png
5th Configuration.png
Alternative IK Solutions for Initial Pose.png

Somasundar Avantsa

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Sep 18, 2018, 1:49:43 AM9/18/18
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 Dear Sir,

Yes,by finding manipulability we  can sort of find the best alternate path from IK Solutions. But how to find all possible paths in the first place, I guess IKCON or IKINE gives only one path ( Is it the one with best manipulability?).

If I have all possible paths, I can find Manipulability for each path and get the best one.

Best regards,
Somasundar

Erik van Oene

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Sep 18, 2018, 2:40:03 PM9/18/18
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Hi,
I was under the impression you have something like 5 of 8 alternatives.

To find all trajectories will be hard as there are infinitely many trajectories.

You have to see ikcon ikine more as a joint position solution to a requested pose that a algorithm to generate joint trajectories.

Erik.

Somasundar Avantsa

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Jul 4, 2020, 7:27:29 AM7/4/20
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To 

Prof. Peter Corke,

Australia


Respected Sir,

 

Greetings of the day!!!

 

Robotic Toolbox has been of immense help to me and your valuable replies to my queries have guided in fact me to execute some basic and 3D complex paths by following Inverse Kinematics and Inverse Jacobian approaches and estimating/ avoiding  singularities. 


Sir, however since the Industrial or now-a-days service robots basically use Cartesian programming to achieve offline/online path planning, I would like learn practical aspects of Robotic path planning.

  

Would you please suggest me to go through any research articles/websites/conferences or softwares to actually relate theoretical approaches like IK/IJ approaches of path planning with Cartesian control/programming used on floor robots or actual use of Jacobian Matrix  and other concepts in Cartesian motion planning.

 

Thanking you,sir,

 

Best regards,

AV Somasundar

Research Scholor

KLEF

Vijayawada

India


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Peter Corke

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Jul 4, 2020, 6:49:19 PM7/4/20
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Check out Robot Academy at robotacademy.net.au, there are relevant master classes (series of lessons)

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