AXY:Basics of joint parameters identification

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.

Mathematical derivation

In a simple case an actuating joint can be described with the following motion equation:

$\tau =\Theta \cdot {\ddot {x}}+\rho _{v}\cdot {\dot {x}}+\rho _{c}\cdot sign({\dot {x}})\qquad \mathrm {(1)}$

Here the symbols are:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau}	Torque (feedback)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dot{x}, \ddot{x}}	Velocity and acceleration (feedback)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Theta}	Moment of inertia
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho_v}	Viscous friction coefficient
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho_c}	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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Theta} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho_v} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho_c} .

The recorded value tuples can be represented as a over-determined linear equation system:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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} }

These n equations can also be written as a single matrix equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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)} }

One solution of this system of equations can be obtained by using the pseudoinverse of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{X}} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{p} = \left ( \mathbf{X}^T \cdot \mathbf{X} \right )^{-1} \cdot \mathbf{X}^T \cdot \boldsymbol{\tau} \qquad \mathrm{(3)} }

HowTo identify joint parameters

1. Use AF function IDM_IdentParamsJoint to move an identification profile and create a record.
2. Process the obtained record with the aico.motionAnalyser function joint_identDynamics.

Verification

See verification page.