By now you're familiar with the equations
of motion of a robot.
In this video we focus on better understanding
the mass matrix M of theta.
First, recall that the kinetic energy of a
point mass is one-half m-v-squared, where
m is the mass and v is its scalar velocity.
If v is a vector, we could rewrite this as
one-half v-transpose times m times v.
Now, for a robot arm, it is not hard to show
that the kinetic energy takes the same form,
one-half theta-dot-transpose times the mass
matrix times theta-dot.
The mass matrix is positive definite, meaning
that the kinetic energy is positive for any
nonzero joint velocity vector.
This is analogous to the fact that a point
mass can only have positive mass.
In addition, the mass matrix is symmetric.
Finally, the mass matrix depends on the joint
configuration theta.
The mass matrix depends on theta because the
amount of inertia about each joint depends
on whether the arm is stretched out or not.
To see the variation in the mass matrix graphically,
consider again the 2R robot arm, where the
link lengths and masses are each one.
Assume that the robot initially has zero velocity,
and consider a circle of accelerations in
the joint space at this robot configuration.
Then this circle maps through the mass matrix
to an ellipse of joint torques.
This ellipse can be interpreted as a direction-dependent
mass ellipsoid; certain joint acceleration
directions require larger torques than others.
The directions of the principal axes of the
ellipse are given by the eigenvectors of the
mass matrix and the lengths of the principal
semi-axes are given by the eigenvalues.
If the mass matrix is invertible, then we
can also map a circle of joint torques to
an ellipse of joint accelerations.
If we change the configuration of the robot,
the shapes of these ellipses change.
Since these ellipses are in joint torque and
acceleration space, they are not easy to understand
intuitively.
Instead, imagine that you grab the endpoint
of the robot and you feel how "massy" it is
when you move it in different directions.
Let's say that V is the endpoint linear velocity,
related to the joint velocity by the Jacobian
J. When you linearly accelerate the endpoint,
you will feel an apparent mass at the end-effector
that depends on the joint configuration.
We call this apparent mass Lambda of theta.
To see how Lambda is related to the mass matrix
M, we can equate the kinetic energy expressed
in the end-effector velocity and the joint
velocity.
If the Jacobian is invertible, we can express
the joint velocity as J-inverse times V, which
gives us the relationship we were looking
for: the configuration-dependent end-effector
mass is equal to J-inverse-transpose times
M times J-inverse.
Now, if you consider a circle of endpoint
accelerations when the robot is at rest, we
can map this through the end-effector mass
Lambda to get an ellipsoid of endpoint forces,
depending on the robot's configuration.
This ellipse is easier to understand.
First of all, the directions of the force
and the endpoint acceleration are only aligned
if the force is aligned with a principal axis
of the ellipse, as you see here.
To accelerate the endpoint in this direction,
you need a lot of force.
To accelerate the endpoint in the orthogonal
direction, you need much less force.
For all force directions not aligned with
a principal axis of the ellipse, the acceleration
direction is not parallel to the force direction.
To see this, let's map a circle of endpoint
forces through the inverse end-effector mass
matrix to get an ellipse of end-effector accelerations.
For an endpoint force purely in the x-direction,
as indicated by the dot on the circle of forces,
we get an end-effector acceleration that has
both x and y components, as indicated by the
dot on the ellipsoid of accelerations.
From this example, we learn two things.
First, the magnitude of the effective end-effector
mass depends on the direction of acceleration.
Second, in general the directions of the end-effector
acceleration and force are not aligned.
So when we move the endpoint of the robot
by hand, it does not feel like a point mass,
which has a constant mass magnitude and always
accelerates in the direction of the applied
force.
Also, the apparent end-effector mass depends
on the configuration of the robot, as you
see here.
You should now have a good understanding of
the form of the dynamic equations of a robot,
including the mass matrix and velocity-product
terms.
Intuitively, these equations of motion are
just f equals m-a, where the m-a term depends
on both the joint velocities and accelerations,
plus forces to balance gravity, plus forces
to create the desired wrench at the end-effector.
Starting in the next video, we will learn
another way to derive these same equations,
beginning with the equation f equals m-a for
a single rigid body.
This is called the Newton-Euler formulation
of the dynamics.
This formulation allows us to derive an efficient
recursive algorithm, without differentiations,
for computing the dynamics of open-chain robots.
