1. 1 Introduction
In linear control, we modeled the manipulator by $n$ independent second-order differential equations.
In this chapter, we will base our controller design on the $n \times 1$ vector differential equation
2. 2 Nonlinear
- local linearization
When nonlinearities are not severe, local linearization can be used in the neighborhood of an operating point.
But, manipulator move among regions so widely separated that no linearization valid for all regions can be found.
- moving linearization
move the operating point with the manipulator as it
moves, always linearizing about the desired position of the manipulator
- deal with the nonlinear directly
the poles of the system will “move”, so we can not select fixed gains. Instead, the gains are also time-varying so it will keep the system critically damped
- Example: nonlinear spring
open loop equation: $m \ddot { x } + b \dot { x } + q x ^ { 3 } = f$
servo portion: $f ^ { \prime } = \ddot { x } _ { d } + k _ { v } \dot { e } + k _ { p } e$
model-based portion: changed
$\begin{array} { l } { \alpha = m } \ { \beta = b \dot { x } + q x ^ { 3 } } \end{array}$
##3 Control problem for Manipulators
For manipulator, the model is complicated
$$
\tau = M ( \Theta ) \ddot { \Theta } + V ( \Theta , \dot { \Theta } ) + G ( \Theta )
$$
where $\Theta $ is the position of all the joints.
If we add friction to the model, we get
$$
\tau = M ( \Theta ) \ddot { \Theta } + V ( \Theta , \dot { \Theta } ) + G ( \Theta ) + F ( \Theta , \dot { \Theta } )
$$
where we can use our partitioned controller again:
$$
\tau = \alpha \tau ^ { \prime } + \beta
$$
and we choose
$$
\begin{aligned} \alpha & = M ( \Theta ) \ \beta & = V ( \Theta , \dot { \Theta } ) + G ( \Theta ) + F ( \Theta , \dot { \Theta } ) \end{aligned}
$$
servo law: $\tau ^ { \prime } = \ddot { \Theta } _ { d } + K _ { v } \dot { E } + K _ { p } E$ where $E = \Theta _ { d } - \Theta$
finally we get
$$
\ddot { E } + K _ { v } \dot { E } + K _ { p } E = 0
$$
This solve the problem in theory, but not in practice because
- computer do it by discrete nature
- inaccuracy in manipulator model
3. 4 Current industrial-robot Control System
Parameters may be inaccurate. So model-based control law maybe doesn’t make sense
For economic reasons, error driven is more usual.
- individual-joint PID control
average gain are chosen
$\tau ^ { \prime } = \ddot { \Theta } _ { d } + K _ { v } \dot { E } + K _ { p } E + K _ { i } \int Edt$
4. 5 Lyapunov Stability Analysis
A analytically way to evluate stability but not the performance.