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Previously, we have seen stiffness control in work space, which was quite obvious in terms of interactions

2
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happen in operational space.

3
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But what happens if we try to apply stiffness control in joint space?

4
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Would that give us additional benefits or it is totally useless and is additional pain?

5
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Let's try to investigate that issue.

6
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First, let's see dynamical model of robot manipulator.

7
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Let's take the input control as PD plus gravity control method, but applied in drawing space.

8
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As you can see, input control is much more easier in joint space than in workspace because of not involving

9
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complex Jacobean terms.

10
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Let's investigate what will happen in steady state conditions, namely, wind velocity and acceleration

11
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terms becomes zero and gravity turns cancel each other.

12
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Don't forget this is not true in practice because we cannot model gravity term.

13
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Exactly.

14
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We can write Equation 1.2 in the form 1.3.

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As we have learned before, we can convert joint corners to workspace coordinates using analytical Jacobean.

16
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Now let's combine equations 1.3 and 1.4 in order to get final equations, which defines that this steady

17
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state error.

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If you analyze Equation one point five, you will see that circle term is nothing but equal in compliance.

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This also gives us definition of stiffness matrix because stiffness is nothing but inverse of compliance.

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This stiffness is configuration dependent because Jacobins are configuration dependent.

21
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So all in all, implementation of stiffness control is easier in joint space because analytical Jacobins

22
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and there are inverse are not involved.

23
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But the stiffness term becomes rather difficult because it changes depending on configuration, which

24
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was not a problem in workspace stiffness control.