Motion Control

We would partition the controller into two parts, model-based portion and servo portion.

The model-based portion makes use of feedback to reduce the system such that it appears like a unit mass.

Control-law decomposition

Given model

\[ m\ddot{x}+b\dot{x}+kx=f, \]

choose \(f=\alpha f'+\beta\) such that

\[ \begin{align} \ddot{x}=f'\quad\text{open-loop}\label{servo-portion} \end{align} \]

so

\[ \alpha=m, \quad \beta=b\dot{x}+kx. \]

we design a control law to compute \(f'\), so

\[ f'=-k_v \dot{x}-k_px \]

the system yields

\[ \ddot{x}+k_v\dot{x}+k_px=0. \]

Trajectory-following control

Given a planned trajectory \(x_d(t)\) is smooth, which means our trajectory generator could give access to \(x_d,\dot{x}_d,\ddot{x}_d\) all the time.

Trajectory following

Define error \(e=x_d-x\), then a servo-control law that cause trajectory following

\[ f'=\ddot{x}_d+k_v\dot{e}+k_pe. \]

combined with equation \(\ref{servo-portion}\), we have

\[ \begin{align} \nonumber \ddot{x}_d+k_v\dot{e}+k_pe&=\ddot{x}\\ \Rightarrow \quad \ddot{e}+k_v\dot{e}+k_pe&=0.\label{error-space} \end{align} \]

which is also called error space.

Disturbance Rejection

Consider a noise \(f_{dist}\), then equation \(\ref{error-space}\) becomes

\[ \ddot{e}+k_v\dot{e}+k_p e=f_{dist}. \]

If \(\max_t f_{dist}(t)<a\), then the consequent error \(e\) is a also bounded.

Modelling & Control of a single joint

We first develop a simplified model of a single rotary joint.

Modelling

For direct current (DC) motor, we have

\[ \tau_m=k_mi_a,\quad V=k_e \dot{\theta}_m. \]

where the first one is a driven torque of the motor, and the second one is the generated voltage of rotation.

In the circuit of the armature, we have

\[ l_a \dot{i}_a+r_ai_a=V_a-V=V_a-k_e\dot{\theta}_m. \]

The gear ratio \(\eta\) causes an increase in the torque seen at the load and a reduction in the speed of the load, given by

\[ \begin{align} \tau=\eta\tau_m,\quad \dot{\theta}=\frac{\dot{\theta}}{\eta}.\label{relation-joint-rotor} \end{align} \]

• Write a dynamic equaiton of the rotor

\[ \tau_m= I_m \ddot{\theta}_m+b_m \dot{\theta}_m+(1/\eta)(I\ddot{\theta}+b\dot{\theta}). \]

where \(I_m\), \(I\) are the inertias of the motor rotor and of the load, respectively, and \(b_m\), \(b\) are viscous friction coefficients for the rotor and load bearings, respectively.

Using equation \(\ref{relation-joint-rotor}\), we write the above equation in terms of motor variables \(\theta_m\) or joint/load variables \(\theta\)

\[ \begin{align*} \tau_m&=\left(I_m+\frac{I}{\eta^2}\right)\ddot{\theta}_m+\left(b_m+\frac{b}{\eta^2}\right)\dot{\theta}_m\\ \tau&=\left(I+I_m \eta^2\right)\ddot{\theta}+\left(b+b_m\eta^2\right)\dot{\theta}. \end{align*} \]

where \(I+I_m \eta^2\) is called the effective inertia seen at the output of the gearing, \(b+b_m\eta^2\) is called the effective damping.