Robot mechanics can experience heavy mechanical oscillations if acceleration excitation frequency comes close to the robots resonance frequency.
To keep oscillations low, excitation frequency of motion generation should be limited to a fraction of resonance frequency.
Stiffness (together with resonance frequency) of the tool centre point of a Delta kinematics significantly varies with its position and excitation direction.
A physical model is needed to predict the resonance frequencies at start and target points of a motion and limit the excitation by adjusting jerks.

== Physical model ==

The platform compliance mainly comprises of two components.
The first follows from the compliance of actuating joints, the second from the compliance of parallelogram rods.

=== Platform stiffness due to actuating joints compliance ===

Here it is assumed that the parallelogram rods are absolutely stiff.
The compliance of the platform only follows from the compliance of the actuating joints.
The method of virtual work is used for the derivation of the stiffness matrix.

The external force <math>\overrightarrow{\mathbf{F}}</math> at the platform position <math>\overrightarrow{\mathbf{P}} = \begin{bmatrix} x & y & z \end{bmatrix}^T</math> causes virtual displacement work that must be equal to the displacement work of actuating joints:
:<math>
\overrightarrow{\mathbf{F}}^T \cdot \Delta \overrightarrow{\mathbf{P}}
- \overrightarrow{\mathbf{M_q}}^T \cdot \Delta \overrightarrow{\mathbf{q}}
= 0
\qquad \mathrm{(1)}
</math>
Vector <math>\overrightarrow{\mathbf{q}}</math> describes the joint positions (angles) and <math>\overrightarrow{\mathbf{M_q}}</math> the joint torques.

Displacement of the actuating joints is described the known Jacobian matrix:
:<math>
\Delta \overrightarrow{\mathbf{q}}
= \mathbf{J} \cdot \Delta \overrightarrow{\mathbf{P}}
\qquad \mathrm{(2)}
</math>

Hooke's law describes the stiffness of the actuating joints:
:<math>
\overrightarrow{\mathbf{M_q}}
= \mathbf{C_q} \cdot \Delta \overrightarrow{\mathbf{q}}
\qquad \mathrm{(3)}
</math>

The diagonal matrix <math>\mathbf{C_q}</math> contains known joint stiffness constants:
:<math>
\mathbf{C_q} =
\begin{bmatrix}
c_q & 0 & 0 \\
0 & c_q & 0 \\
0 & 0 & c_q
\end{bmatrix}
</math>

Inserting (2) in (1) leads to:
:<math>
\left ( \overrightarrow{\mathbf{F}}^T - \overrightarrow{\mathbf{M_q}}^T \cdot \mathbf{J} \right )
\cdot \Delta \overrightarrow{\mathbf{P}}
= 0
\qquad \Rightarrow
\overrightarrow{\mathbf{F}}^T - \overrightarrow{\mathbf{M_q}}^T \cdot \mathbf{J}
= 0
\qquad \mathrm{(4)}
</math>

Hooke's law for the Cartesian stiffness is obtained by inserting (3) and (2) in (4):
:<math>
\begin{array}{lcl}
\overrightarrow{\mathbf{F}}
& = & \mathbf{J}^T \cdot \overrightarrow{\mathbf{M_q}} \\
& = & \mathbf{J}^T \cdot \mathbf{C_q} \cdot \Delta \overrightarrow{\mathbf{q}} \\
& = & \mathbf{J}^T \cdot \mathbf{C_q} \cdot \mathbf{J} \cdot \Delta \overrightarrow{\mathbf{P}} \\
& = & \mathbf{K_q} \cdot \Delta \overrightarrow{\mathbf{P}}
\end{array}
\qquad \mathrm{(5)}
</math>

Matrix <math>\mathbf{K_q}</math> is the Cartesian stiffness matrix:
:<math>
\mathbf{K_q} = \mathbf{J}^T \cdot \mathbf{C_q} \cdot \mathbf{J}
</math>

Platform stiffness in direction of the external force is then:
:<math>
k_q
= \frac{\left | \overrightarrow{\mathbf{F}} \right |} {\left |\Delta \overrightarrow{\mathbf{P}} \right |}
= \frac{\left | \mathbf{K_q} \cdot \overrightarrow{\mathbf{P}} \right |} {\left |\Delta \overrightarrow{\mathbf{P}} \right |}
\qquad \mathrm{(6)}
</math>

=== Platform stiffness due to parallelogram rods compliance ===

In this case it is assumed that the actuating joints are absolutely stiff.
The compliance of the platform only results from parallelogram rod compliance.
Method of virtual work is again used for the derivation.

The external force <math>\overrightarrow{\mathbf{F}}</math> at the platform position <math>\overrightarrow{\mathbf{P}} = \begin{bmatrix} x & y & z \end{bmatrix}^T</math> causes counter forces in the parallelogram rods <math>F_{c,i}</math>.
[[Rod forces model for Delta kinematics]] describes the computation of these rod forces.
Forces can be treated as scalar values because all forces directions match the displacement directions.

The (known) stiffness of the parallelogram rods is then:
:<math>
c_r
= \frac {F_{c,i}} {\Delta l_{b,i}}
\qquad \mathrm{(7)}
</math>
Factor <math>l_{b,i}</math> is the length of the rod.

The external force at the platform position <math>\overrightarrow{\mathbf{P}} = \begin{bmatrix} x & y & z \end{bmatrix}^T</math> causes virtual displacement work that must be equal to the displacement work of the parallelogram rods:
:<math>
F \cdot \Delta P
- \sum_{i} F_{c,i} \cdot \Delta l_{b,i}
= 0
\qquad \mathrm{(8)}
</math>

Inserting (7) in (8) and rearranging leads to:
:<math>
F \cdot \Delta P
- \sum_{i} \frac {F_{c,i}^2} {c_r}
= 0
\qquad \Rightarrow
\Delta P
= \frac {\sum_{i} \frac {F_{c,i}^2} {c_r}} {F}
\qquad \mathrm{(9)}
</math>

The Cartesian stiffness of the platform is obtained by invoking Hooke's law:
:<math>
k_r
= \frac {F} {\Delta P}
= \frac {F^2} {\sum_{i} \frac {F_{c,i}^2} {c_r}}
= \frac {F^2} {\sum_{i} F_{c,i}^2} \cdot c_r
\qquad \mathrm{(10)}
</math>

=== Total stiffness of the platform ===

Total stiffness of the platform is a serial set-up of <math>k_q</math> and <math>k_r</math>:
:<math>
k
= \frac {k_q \cdot k_r} {k_q + k_r}
\qquad \mathrm{(11)}
</math>

=== Platform resonance frequency ===

The resonance frequency is dependent of the system's mass and the spring constant.
In case of Delta robot the mass consists of the platform mass and 1/3 of the parallelogram rod masses:
:<math>
m
= m_{platform} + 3 \cdot \left ( \frac{1}{3} \cdot m_{BC} \right )
= m_{platform} + m_{BC}
</math>

The platform resonance frequency is then:
:<math>
f_{res}
= \frac {\sqrt{\frac{k}{m_{platform} + m_{BC}}}} {2 \pi}
\qquad \mathrm{(12)}
</math>

== Limitation of excitation frequency ==

The maximal excitation frequency is calculated as rate of resonance frequency:
:<math>
F_{exc} = R_{ext} \cdot F_{res}
</math>

From that relation follow the minimal acceleration/deceleration time:
:<math>
T_{AccMin} = \frac{1}{2} \cdot \frac{1}{F_{exc}}
</math>

== Parameters ==

The both model parameters <math>c_q</math> and <math>c_r</math> are obtained by estimation.
It is helpful to measure the resonance frequencies at different TCP positions and different excitation direction with an accelerometer and manually adjust the parameters.

The excitation frequency rate <math>R_{ext}</math> is adjusted by optimizing the total time of motions.
Lower values will reduce settling time but prolong the commanded motion.

== Verification ==

See [[/Verification|verification page]].

== See also ==

* In [[aico.motionAnalyser]] this model is implemented by the function [[delta_resonanceFrequency]].

[[Category:Model based control for Delta kinematics]]

AXY:Resonance frequency model for Delta kinematics

2012-12-24T13:20:50Z

Arwiebe: Arwiebe moved page CNT:Resonance frequency model for Delta kinematics to AXY:Resonance frequency model for Delta kinematics

AXY:Rod forces model for Delta kinematics

2012-12-24T13:20:34Z

Arwiebe: Arwiebe moved page CNT:Rod forces model for Delta kinematics to AXY:Rod forces model for Delta kinematics

AXY:Basics of joint parameters identification

2012-12-24T13:17:51Z

Arwiebe:

Model based control can bring great benefit to robot control, but it requires exact model parameters.
This document describes how these parameters can be identified (automatically).

== Record identification motion profile ==

The motion profile should provoke all motion states described by the model.
Ideally it should move over the whole position range of the joint, have plenty accelerations/decelerations and have different constant velocity phases.

The recorded file must contain the position feedback along with the torque feedback.

== Derivative without phase shift ==

For parameter identification joint velocity and acceleration is needed.
Standard numeric derivative of position feedback leads to phase shift, which impairs the identification.
To avoid the phase shift interpolated derivative should be used instead.
Hereby the numeric derivative is spline-interpolated and values at original sampling times are taken.

Interpolated derivative is implemented in [[Control:aico.motionAnalyzer|aico.motionAnalyser]] by the function [[signal_deriveInterp]].

== Mathematical derivation ==

In a simple case an actuating joint can be described with the following motion equation:
:<math>
\tau = \Theta \cdot \ddot{x} + \rho_v \cdot \dot{x} + \rho_c \cdot sign(\dot{x})
\qquad \mathrm{(1)}
</math>

Here the symbols are:
:{|
|<math>\tau</math>
|Torque (feedback)
|-
|<math>\dot{x}, \ddot{x}</math>
|Velocity and acceleration (feedback)
|-
|<math>\Theta</math>
|Moment of inertia
|-
|<math>\rho_v</math>
|Viscous friction coefficient
|-
|<math>\rho_c</math>
|Coulomb friction coefficient
|}

It is assumed that the joints position, velocity and acceleration feedbacks together with the torque feedback are known (recorded).
The unknown model parameters are <math>\Theta</math>, <math>\rho_v</math> and <math>\rho_c</math>.

The recorded value tuples can be represented as a over-determined linear equation system:
:<math>
\begin{array}{lcl}
\tau_1 & = & \Theta \cdot \ddot{x}_1 + \rho_v \cdot \dot{x}_1 + \rho_c \cdot sign(\dot{x}_1) \\
\tau_2 & = & \Theta \cdot \ddot{x}_2 + \rho_v \cdot \dot{x}_2 + \rho_c \cdot sign(\dot{x}_2) \\
\tau_3 & = & \Theta \cdot \ddot{x}_3 + \rho_v \cdot \dot{x}_3 + \rho_c \cdot sign(\dot{x}_3) \\
& & \vdots \\
\tau_n & = & \Theta \cdot \ddot{x}_n + \rho_v \cdot \dot{x}_n + \rho_c \cdot sign(\dot{x}_n) \\
\end{array}
</math>

These n equations can also be written as a single matrix equation:
:<math>
\begin{bmatrix}
\tau_1 \\
\tau_2 \\
\tau_3 \\
\vdots \\
\tau_n
\end{bmatrix}
=
\begin{bmatrix}
\ddot{x}_1 & \dot{x}_1 & sign(\dot{x}_1) \\
\ddot{x}_2 & \dot{x}_2 & sign(\dot{x}_2) \\
\ddot{x}_3 & \dot{x}_3 & sign(\dot{x}_3) \\
\vdots & \vdots & \vdots \\
\ddot{x}_n & \dot{x}_n & sign(\dot{x}_n) \\
\end{bmatrix}
\cdot
\begin{bmatrix}
\Theta \\
\rho_v \\
\rho_c \\
\end{bmatrix}
\qquad \Rightarrow
\boldsymbol{\tau} = \mathbf{X} \cdot \mathbf{p}
\qquad \mathrm{(2)}
</math>

One solution of this system of equations can be obtained by using the pseudoinverse of <math>\mathbf{X}</math>.
:<math>
\mathbf{p} = \left ( \mathbf{X}^T \cdot \mathbf{X} \right )^{-1} \cdot \mathbf{X}^T \cdot \boldsymbol{\tau}
\qquad \mathrm{(3)}
</math>

== HowTo identify joint parameters ==

# Use AF function [[CNT:AF:IDM IdentParamsJoint|IDM_IdentParamsJoint]] to move an identification profile and create a record.
# Process the obtained record with the [[Control:aico.motionAnalyzer|aico.motionAnalyser]] function ''joint_identDynamics''.

== Verification ==

See [[/Verification|verification page]].

[[Category:Axystems:Motion Dynamics]]

AXY:Basics of joint parameters identification/Verification

2012-12-24T13:17:19Z

Arwiebe: Arwiebe moved page CNT:Basics of joint parameters identification/Verification to AXY:Basics of joint parameters identification/Verification

[[File:Basics of joint parameters identification - Verification.png|1000px]]

The image shows a comparison between recorded torque feedback and torque computed using the identified parameters.

== Repeatability of parameter identification ==

For testing the repeatability of implemented parameter identification algorithm the parameters of joint 2 (DR1200-4S) are identified several times consecutively.
At the beginning of all measurements the joint is already in operating temperature.

{|border="1" cellpadding="5"
|'''Measurement '''
|<math>\Theta_{AB}</math>
|<math>\rho_v</math>
|<math>\rho_c</math>
|<math>l_{AB}m_{AB}g</math>
|-
|1
|0,1344732
|0,3389793
|3,8477142
|2,0104706
|-
|2
|0,1345419
|0,3383904
|3,8244827
|2,0297676
|-
|3
|0,1345093
|0,3334403
|3,8569023
|2,0208896
|-
|4
|0,1345672
|0,3398903
|3,7731432
|2,0273873
|-
|5
|0,1345354
|0,3346488
|3,7522782
|2,0209898
|-
|6
|0,1345903
|0,3314771
|3,7464159
|2,0335845
|-
|7
|0,1346014
|0,3246576
|3,7571930
|2,0183868
|-
|8
|0,1346407
|0,3281108
|3,7361262
|2,0274325
|-
|9
|0,1345969
|0,3239006
|3,7094229
|2,0246346
|-
|10
|0,1346543
|0,3223137
|3,7282359
|2,0401946
|-
|'''Average'''
|0,1345711
|0,3315809
|3,7731915
|2,0253738
|-
|'''Mean variation'''
|'''0,13%'''
|'''5,30%'''
|'''3,91%'''
|'''1,47%'''
|}

== Conclusions ==

As can be seen in the table all parameters can be reproducibly identified.

AXY:Basics of joint parameters identification

2012-12-24T13:17:18Z

Arwiebe: Arwiebe moved page CNT:Basics of joint parameters identification to AXY:Basics of joint parameters identification

MediaWiki:Namespaces-admin

2012-12-24T12:28:52Z

Arwiebe:

MediaWiki:Namespaces-admin

2012-12-24T11:46:48Z

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2012-12-23T11:54:55Z

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2012-11-11T08:10:20Z

Arwiebe:

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2012-11-11T08:09:18Z

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