AXY:Resonance frequency model for Delta kinematics

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 ${\overrightarrow {\mathbf {F} }}$ at the platform position ${\overrightarrow {\mathbf {P} }}={\begin{bmatrix}x&y&z\end{bmatrix}}^{T}$ causes virtual displacement work that must be equal to the displacement work of actuating joints:

${\overrightarrow {\mathbf {F} }}^{T}\cdot \Delta {\overrightarrow {\mathbf {P} }}-{\overrightarrow {\mathbf {M_{q}} }}^{T}\cdot \Delta {\overrightarrow {\mathbf {q} }}=0\qquad \mathrm {(1)}$

Vector ${\overrightarrow {\mathbf {q} }}$ describes the joint positions (angles) and ${\overrightarrow {\mathbf {M_{q}} }}$ the joint torques.

Displacement of the actuating joints is described the known Jacobian matrix:

$\Delta {\overrightarrow {\mathbf {q} }}=\mathbf {J} \cdot \Delta {\overrightarrow {\mathbf {P} }}\qquad \mathrm {(2)}$

Hooke's law describes the stiffness of the actuating joints:

${\overrightarrow {\mathbf {M_{q}} }}=\mathbf {C_{q}} \cdot \Delta {\overrightarrow {\mathbf {q} }}\qquad \mathrm {(3)}$

The diagonal matrix $\mathbf {C_{q}}$ contains known joint stiffness constants:

$\mathbf {C_{q}} ={\begin{bmatrix}c_{q}&0&0\\0&c_{q}&0\\0&0&c_{q}\end{bmatrix}}$

Inserting (2) in (1) leads to:

$\left({\overrightarrow {\mathbf {F} }}^{T}-{\overrightarrow {\mathbf {M_{q}} }}^{T}\cdot \mathbf {J} \right)\cdot \Delta {\overrightarrow {\mathbf {P} }}=0\qquad \Rightarrow {\overrightarrow {\mathbf {F} }}^{T}-{\overrightarrow {\mathbf {M_{q}} }}^{T}\cdot \mathbf {J} =0\qquad \mathrm {(4)}$

Hooke's law for the Cartesian stiffness is obtained by inserting (3) and (2) in (4):

${\begin{array}{lcl}{\overrightarrow {\mathbf {F} }}&=&\mathbf {J} ^{T}\cdot {\overrightarrow {\mathbf {M_{q}} }}\\&=&\mathbf {J} ^{T}\cdot \mathbf {C_{q}} \cdot \Delta {\overrightarrow {\mathbf {q} }}\\&=&\mathbf {J} ^{T}\cdot \mathbf {C_{q}} \cdot \mathbf {J} \cdot \Delta {\overrightarrow {\mathbf {P} }}\\&=&\mathbf {K_{q}} \cdot \Delta {\overrightarrow {\mathbf {P} }}\end{array}}\qquad \mathrm {(5)}$

Matrix $\mathbf {K_{q}}$ is the Cartesian stiffness matrix:

$\mathbf {K_{q}} =\mathbf {J} ^{T}\cdot \mathbf {C_{q}} \cdot \mathbf {J}$

Platform stiffness in direction of the external force is then:

$k_{q}={\frac {\left|{\overrightarrow {\mathbf {F} }}\right|}{\left|\Delta {\overrightarrow {\mathbf {P} }}\right|}}={\frac {\left|\mathbf {K_{q}} \cdot {\overrightarrow {\mathbf {P} }}\right|}{\left|\Delta {\overrightarrow {\mathbf {P} }}\right|}}\qquad \mathrm {(6)}$

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 ${\overrightarrow {\mathbf {F} }}$ at the platform position ${\overrightarrow {\mathbf {P} }}={\begin{bmatrix}x&y&z\end{bmatrix}}^{T}$ causes counter forces in the parallelogram rods $F_{c,i}$. 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:

$c_{r}={\frac {F_{c,i}}{\Delta l_{b,i}}}\qquad \mathrm {(7)}$

Factor $l_{b,i}$ is the length of the rod.

The external force at the platform position ${\overrightarrow {\mathbf {P} }}={\begin{bmatrix}x&y&z\end{bmatrix}}^{T}$ causes virtual displacement work that must be equal to the displacement work of the parallelogram rods:

$F\cdot \Delta P-\sum _{i}F_{c,i}\cdot \Delta l_{b,i}=0\qquad \mathrm {(8)}$

Inserting (7) in (8) and rearranging leads to:

$F\cdot \Delta P-\sum _{i}{\frac {F_{c,i}^{2}}{c_{r}}}=0\qquad \Rightarrow \Delta P={\frac {\sum _{i}{\frac {F_{c,i}^{2}}{c_{r}}}}{F}}\qquad \mathrm {(9)}$

The Cartesian stiffness of the platform is obtained by invoking Hooke's law:

$k_{r}={\frac {F}{\Delta P}}={\frac {F^{2}}{\sum _{i}{\frac {F_{c,i}^{2}}{c_{r}}}}}={\frac {F^{2}}{\sum _{i}F_{c,i}^{2}}}\cdot c_{r}\qquad \mathrm {(10)}$

Total stiffness of the platform

Total stiffness of the platform is a serial set-up of $k_{q}$ and $k_{r}$:

$k={\frac {k_{q}\cdot k_{r}}{k_{q}+k_{r}}}\qquad \mathrm {(11)}$

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:

$m=m_{platform}+3\cdot \left({\frac {1}{3}}\cdot m_{BC}\right)=m_{platform}+m_{BC}$

The platform resonance frequency is then:

$f_{res}={\frac {\sqrt {\frac {k}{m_{platform}+m_{BC}}}}{2\pi }}\qquad \mathrm {(12)}$

Limitation of excitation frequency

The maximal excitation frequency is calculated as rate of resonance frequency:

$F_{exc}=R_{ext}\cdot F_{res}$

From that relation follow the minimal acceleration/deceleration time:

$T_{AccMin}={\frac {1}{2}}\cdot {\frac {1}{F_{exc}}}$

Parameters

The both model parameters $c_{q}$ and $c_{r}$ 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 $R_{ext}$ is adjusted by optimizing the total time of motions. Lower values will reduce settling time but prolong the commanded motion.

Verification

See verification page.

See also

• In aico.motionAnalyser this model is implemented by the function delta_resonanceFrequency.