We've been assuming that there is an actuator at each joint that directly creates the joint
force or torque.
This is the idea behind robots called direct-drive robots: there is a motor at each joint that
creates torque without any gearing.
Such designs are often impractical, though, since an electric motor with the right power
rating often spins at high speed and low torque, whereas most robotic applications require
high torque.
In practice, robots use many different types of actuators, such as electric motors, hydraulic
actuators, and pneumatic actuators, and different types of mechanical power transformers and
transmissions, such as timing belts and pulleys, chains and sprockets, cables, and gear trains.
Each combination of actuator and transmission has its own dynamic characteristics that should
be taken into account in the robot dynamics.
In this final video of Chapter 8, we consider one popular choice for actuation: an electric
motor with a gearhead located at each joint.
The gearhead increases the torque of the motor while reducing the speed.
This is an image of a typical robot actuator.
At one end of the motor is an encoder, a sensor that measures how far the motor has rotated,
so we know the joint position.
The motor itself consists of a stator, the portion we think of as remaining stationary,
and the rotor, the portion that rotates relative to the stator.
The rotor includes the motor shaft, while the stator includes the motor housing.
Because this particular electric motor is a brushed motor, where current is carried
to the motor coils, or windings, through brushes sliding on a commutator, the windings are
part of the rotor and the magnets are part of the stator.
For brushless motors, which are more commonly used in robots, the windings are part of the
stator and the magnets are part of the rotor.
Both brushed and brushless motors generate torque by sending a current through windings
in a magnetic field created by magnets.
Because the motor spins at high speed, often up to 10,000 revolutions per minute, but low
torque, it is attached to a gearhead.
It is the output shaft of the gearhead that spins the next link.
An ideal gearhead decreases the speed by the gear ratio G, where G is greater than 1, and
increases the torque by the factor G.
This preserves the power of the motor while transforming the motor's output to more useful
high torques and low speeds.
In practice, the torque amplification is somewhat less than G, due to friction, gear-teeth impact,
and other power losses in the gearhead.
This figure shows how a geared motor is typically used in a robot joint.
The stator is attached to link i-minus-1 and the gearhead output shaft is attached to link
i.
The mass and inertia of the stator should be counted as part of link i-minus-1, while
the mass and inertia of the rotor should be counted as part of link i.
It is not quite this simple, though, because the axis of the joint, which is aligned with
the gearhead axis, may not be the same as the rotor axis, and because the rotor spins
at a different speed than the joint.
Typically the mass and inertia of a motor's rotor is much less than the mass and inertia
of link i, so it's tempting to ignore the rotor's mass and inertia.
But the rotor spins G times faster than link i because of the gearhead, so the effect of
the motor's inertia could be significant.
To see this, we can calculate the kinetic energy of the rotor as one-half the scalar
inertia of the rotor about its rotational axis times the square of G theta-dot, where
theta-dot is the joint velocity.
This means that the apparent inertia of the rotor about its axis is G-squared times I_rotor.
This is called the apparent inertia since someone manually moving joint i would feel
this apparent rotor inertia, in addition to the inertia of the link.
So even though I_rotor may be small compared to the inertia of the link about the joint
axis, the apparent rotor inertia G-squared times I_rotor may not be small, especially
considering that gear ratios of one hundred or more are common.
Therefore the rotor inertia should be included in our dynamic analysis.
As an example, consider a 2R robot arm with a geared motor at each joint.
For a particular choice of link lengths, masses, and rotor inertias, a gear ratio of ten at
the gearheads yields this mass matrix for the robot.
If we keep everything else the same but increase the gear ratio to one hundred, we get this
mass matrix.
The mass matrix becomes much larger along the diagonal due to the increased apparent
inertia of the rotors.
The off-diagonal elements of the mass matrix are now relatively small compared to the diagonal
elements, and the amount the mass matrix varies with configuration is now relatively small.
This means that the velocity-product terms in the dynamics are comparatively less significant.
As the gear ratios become large, the apparent inertias of the rotors dominate the dynamics,
and the coupled dynamics of the robot become closer and closer to the dynamics of n independent
joints.
Taking into account gearing at each joint, we can derive a modified recursive Newton-Euler
inverse dynamics algorithm.
The details are left to the book, but essentially we calculate the motor torque needed to accelerate
both the rotor and the link.
For electric motors, the torque is proportional to the current through the motor, so for each
motor the robot controller could command a current proportional to the calculated torque.
Finally, we can add an estimate of joint friction torques.
Commonly the amount of joint friction increases with increasing gear ratios.
Some simple models of joint friction are discussed in the book.
This concludes Chapter 8.
Like Chapter 3, which establishes key concepts in spatial motion with applications throughout
the rest of the book, Chapter 8 establishes key concepts in dynamics, with applications
in simulation, robot control in Chapter 11, and planning of minimum-time robot trajectories
in the next chapter, Chapter 9.
Kevin LynchProfessor
