8.1.3. Understanding the Mass Matrix

This video interprets the mass matrix of a robot in terms of how a sphere of joint torques maps to an ellipsoid of joint accelerations and vice-versa, and how a sphere of end-effector wrenches maps to an ellipsoid of end-effector accelerations and vice-versa.

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.