Dynamic modelling of robots: parameters identification (II)

Hi! In this post I will explain the method for obtaining a set of equations to be able to identify the dynamic model of robot composed by several links even in presence of closed loops. In the previous post I detailed how to obtain an expression of the forces and torques acting on the last link as a linear function of the robot parameters. Now let us include the rest of the links.

We said that the spatial force acting on that robotic link is:

f_n = A_n \cdot \phi_n     (1)

However, here the interest lays on identifying the inertial and mass parameters of the whole robotic structure and the effect of all links should be taken into account.

Defining f_ij  as the spatial force at joint i due to movement of j  alone. Then f_ii  is the spatial force at i due to its own movement. The equivalent expression for the (1) would thus be (2) where n  it has been substituted by i . And the superscript i  has been added to indicate that the vectors are expressed in terms of joint  i .

{^i}f_{ii} = {^i}A_i \cdot \phi_i     (2)

In this way, the total spatial force at joint,{^i}f_{i} , it is the sum of the spatial forces $latex{^i}f_{ij}$ for all links  distal to  f_ii , following the results of:

{^i}f_{i} = \sum{^n_{j=1}} {^i}f_{ij}    (3)

The spatial force f_ij  at joint i  its determined by the spatial force transformation matrix {^i}X^F_{i} = ,

{^i}f_{i,i+1} = {^i}X^F_{i+1} \cdot ^{i+1}f_{i+1, i+1}   (4)

So the forces and torques on link i due to the movements of the link  j  can be obtained by cascading a series of transformation matrices:

{^i}f_{i,j} = {^i}X^F_{i+1}\cdot ^{i+1}X^F_{i+2}  \cdots ^{j-1}X^F_{j}\cdot ^{i+1}f_{j, j} ={^i}X^{Fj}_{j} \cdot A_j \cdot \phi_j      (5)

With this approach, the spatial forces of a serial chain robot can be easily expressed in a matrix:

$latex \left( \begin{array} {c} {^1}f_1 \\  {^2}f_2 \\  \vdots \\ {^n}f_n \end{array} \right)  =\begin{bmatrix}
{^1}X_{1}^{F1}A_1 & {^1}X_{2}^{F2} A_2 & \dots & {^1}X_{n}^{Fn}A_n \\
0 & {^2}X_{2} ^{F2}A_2& \dots & {^2}X_{n} ^{Fn}A_n\\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0  & \dots & {^n}X_{n}^{Fn}A_n

\left( \begin{array} {c} \phi_1 \\  \phi_2 \\  \vdots  \\  \phi_n \end{array} \right) $  (6)

Or in compact form: f=A \cdot \phi_i   Since only the torque can be usually measured around the axis z_i , the spatial force can be projected around that rotation axis and simplifying (6) to:

\tau=K \cdot \phi_i        (7)


\tau_i =\left( \begin{array} {c} z_i \\  0  \end{array} \right) \cdot f_i  

K_{ij} =\left( \begin{array} {c} z_i \\  0  \end{array} \right) \cdot {^i}X_{j}^{Fj} \cdot A_j 

\phi==\left( \begin{array} {c} \phi_1 \\  \phi_2 \\  \vdots  \\  \phi_n \end{array} \right)

However, for the very common manipulator with a closed loop such the one presented in before, equation (6) has to be adapted with the inclusion of the closed loop. In order to perform this adaptation, firstly (6) is modified so to include the spanning tree joints as the graph of Figure 1. That leads to an expression similar to  for the matrix  when a non-serial manipulator is modelled.

Kraft GRIPS kinematic model with the reference systems.
Kraft GRIPS kinematic model with the reference systems.

Once the reference systems have been established, following for example the D-H rules, we will be in the position of modifying equation (8) to include the closed loop. This will be explained in detail in the next posts together with the algebra of spatial forces. Be careful with the D-H parameters since there are multiple versions.

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Posted in Robot Dynamics, Robotics

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