Decentralized Joint Control#
The equation of motion of a manipulator without end-effector contact force and any joint friction is
Let
where
Substituting ([equ.motor_angle]{reference-type=”ref” reference=”equ.motor_angle”}) and ([equ.motor_torque]{reference-type=”ref” reference=”equ.motor_torque”}) into the manipulator dynamics ([equ.dyn]{reference-type=”ref” reference=”equ.dyn”}) leads to
We have previously analyzed elements in matrix
where
The above equation ([equ.decouple_model]{reference-type=”ref” reference=”equ.decouple_model”}) will lead to the following diagram.
As shown in Fig. 1{reference-type=”ref” reference=”fig:3”} and
([equ.decouple_model]{reference-type=”ref”
reference=”equ.decouple_model”}), if we don’t consider
Note that in the above equation
for each joint motor and we can
control each joint individually. However,
Single-Joint Control Diagram#
For each joint (we omit the joint index
Here,
The above ([equ.motor]{reference-type=”ref” reference=”equ.motor”}) corresponds to the following diagram:
In the above diagram, the closed loop is due to the DC motor model
itself. We write input (
Minimal Control Background
For ease of the reader, we provide the following minimal background of
control design. A typical closed-loop control diagram is shown below.
Here,
For the above control system, the forward path transfer function is
The backward path transfer function is $
The input-to-output transfer function is
The disturbance-to-output transfer function is
The input (
Background of final value theorem
If a continuous signal
Single Joint Position feedback#
The single joint motor system in Fig
2{reference-type=”ref”
reference=”fig:motor_model2”} is the plant we want to control. The plant
transfer function
We want to design a proportional-integral controller
with
The control diagram thus is shown below (note that the backward pass
constant
The transfer function of the forward path is
The transfer function of the backward pass is
The root locus can be plotted as a function of the gain of the position
loop
If
, the root locus is shown as below. Thus, the system is inherently unstable.If
, the root locus is shown as below. As increases, the absolute value of the real part of the two roots of the locus tending towards the asymptotes increases too, and the system has faster time response.
The closed-loop input/output transfer function is
and the characteristic equation can be factorized into the following form
where
The closed-loop disturbance/output transfer function is
If
thus,
Hence, the controller can cancel the effect of constant disturbance, and the quantity
can be interpreted as the disturbance rejection factor for velocity (or
higher-order) disturbance. Increasing
Single-Joint Position and Velocity Feedback#
To control the motor system in Fig.
2{reference-type=”ref”
reference=”fig:motor_model2”}, we use both position and velocity
controller $
The control diagram is shown below (Note that
which can be further reduced into the following diagram:
The transfer function of the forward path is
The transfer function of the backward path is
For simplicity, one can design
then that the poles of the
closed-loop system move on the root locus as a function of the loop gain
By increasing the position gain
The closed-loop input/output transfer function is
It can be recognized that, with a suitable choice of the gains, it is
possible to obtain any value of natural frequency
For given transducer constants
which shows that the disturbance rejection factor is
If
thus,
Hence, the controller can cancel the effect of constant disturbance, and the quantity
can be interpreted as the disturbance rejection factor for velocity (or
higher-order) disturbance. Increasing