Holonomic constraints

Motivation

In manufacturing systems (i.e. automation and machining), fast response and high precision motion control is an indispensable method such that contouring control, which has been successful for such multi-axis motion control, is addressed.

It is important to point out a constraint, however, most of the existing contouring control approaches have only referred to the designs of controller for the un-constraint motion control systems. Nevertheless, the mechanism of multi-axis machines (e.g., dual-arm robot manipulators) commonly exists the constraint motion systems, especially, contouring control in machining system (e.g., the end-effector of robot have to hold the workpiece to follow a desired path for cutting tool). As a result, joint coordinates of the dual-arm robot manipulators need to satisfy the holonomic constraints, which is formed by algebraic equations [1].

The dual-arm Robot Manipulators

Integration of machine tool and robot: Integration the functions of machining and automation
Integration of serial and parallel robots: With better stiffness and keep the flexibility
Require the motion to follow the specific path
Require the motion to follow the specific surface

Dynamics of the robot with Holonomic constraints

Dynamics of system: \(\ddot{\mathbf{q}}=f(\dot{\mathbf{q}},\mathbf{q})+g(\dot{\mathbf{q}},\mathbf{q})u\)
Holonomic constraints: \(\phi(\mathbf{q})=0\)
\(n+m\) generized coordinates: \(\mathbf{q}\in\mathbb{R}^{n+m}\)
\(n\) actuating inputs: \(u\in\mathbb{R}^n\)
\(m\) constraint functions: \(\phi(\mathbf{q}): \mathbb{R}^{n+m}\to\mathbb{R}^{m}\)
Desired path: \(\mathcal{P}_w(\mathbf{q})=0; \mathbb{R}^{n+m}\to\mathbb{R}^{n-1}\)

Contouring control problem is to seek a control law, \(u\), so that the output \(\mathbf{q}\) follow the desired path \(\mathcal{P}_w\). The main purpose: design such the control law as if there were no constraints using method of \(\textbf{equivalent errors (EQs)}\)

Coordinate transformation framework

A crucial problem of Path-contouring/following Control is to minimize the contour error defined as the shortest distance between the actual displacement \(q(t)\in\mathbb{R}^n\) and the desired path \(\mathcal{P}_w\); \(\epsilon_c(t)=\mathsf{dist}(q,\mathcal{P}_w(q_d))\equiv \displaystyle \|q-q_d\|_{\infty}, \forall{q_d}\in\mathcal{P}_w\). To aim at the new coordinate framework of dynamic model, EQs is addressed by utilizing the \(n−1\) equivalent contour errors determining the desired path in terms of algebraic equation. It becomes the problem of stabilization of Path-contouring Control.

\[ \begin{bmatrix}\epsilon \\e\end{bmatrix}= \begin{bmatrix}\mathcal{P}_w(q) \\\dot{q}_d^T(q-q_d)\end{bmatrix} \]

The existing control techniques can be considered to deal with such the problem of stabilization. It is more practical to develop controller to deal with minimizing contour error with EQ approach than conventional control framework, e.g., a time-dependent tracking control with point-stabilization.

For conventional contouring control:

\[{\epsilon_c(t)\to0 \iff q(t)\to q_d(t), t \to\infty}\]

For contouring control based EQ:

\[{\epsilon_c(t)\to0 \iff \epsilon(t) \to 0, t \to \infty}\]

The new Control objective of contouring Control [2]

Dynamics of errors

\[ \begin{bmatrix}\ddot{\epsilon} \\ \ddot{e}\end{bmatrix}= \Omega(\mathbf{q},\dot{\mathbf{q}},t)+\Gamma(\mathbf{q},\dot{\mathbf{q}},t)\tau \]

\[ {\Omega\in\mathbb{R}^{5\times{1}}}= \left[ \begin{array}{ccc} \mathbf{\mathcal{A}}(\mathbf{q})f(\mathbf{q},\dot{\mathbf{q}})+\mathbf{\mathcal{B}}(\mathbf{q},\dot{\mathbf{q}}) \\ \ddot{\mathbf{q}}_d^T(\mathbf{q}-\mathbf{q}_{d})+\ddot{\mathbf{q}}_{d}^T(\dot{\mathbf{q}}-\dot{\mathbf{q}}_{d})+\dot{\mathbf{q}}_{d}^Tf(\mathbf{q},\dot{\mathbf{q}}) \end{array} \right] \]

\[ {\Gamma\in\mathbb{R}^{5\times{5}}}=\left[ \begin{array}{ccc} {\mathbf{\mathcal{A}}}(\mathbf{q}) \\ \dot{\mathbf{q}}_d^T \end{array} \right]G(\mathbf{q},\dot{\mathbf{q}}) \]

\(\mathcal{A}(\mathbf{q})=[\nabla\mathcal{P}_w(\mathbf{q}_L)\quad0]\) for left-arm

\(\mathcal{A}(\mathbf{q})=[0\quad\nabla\mathcal{P}_w(\mathbf{q}_R)]\) for right-arm

To aim \(\epsilon \to 0\), asymptotically, consider \(\mathbf{{\tau}}\) as feedback linearization

\[ \mathbf{u}=\mathbf{{\tau}}\in\mathbb{R}^{5\times1}={\Gamma}^{-1}(-{\Omega}+v) \]

with control law \(v\) to stabilize the equivalent errors.

Reference

[1] T. Nuchkrua, S. Chang and S. Chen, \(''\)Contouring Control of 5-DOF Manipulator Robot Arm based on Equivalent Errors,\(''\) 2015 International Automatic Control Conference (CACS), Taiwan, 2015.

[2] Y. Dong, T. Nuchkrua and T. Shen, \(''\)Asymptotical Stability Contouring Control of Dual-arm robot with Holonomic Constraints: Modified Distributed Control Framework,\(''\) IET Control Theory \(\&\) Applications, vol. 13, no. 17, pp. 2877-2885, 2019.