Rigid Body Dynamics

The last term we're going to look at is the term 'C' and it represents the Coriolis and Centripetal torques that exists within a serial link robot manipulator. It's often called the velocity term because the C matrix multiplies the vector Q dot, the vector of joint velocities.

This is the torque that is due to gyroscopic effects due to rotating joints within the robot manipulator. There are actually two effects that are modelled by the matrix C and it's the Coriolis effect which is named after the French scientist, mathematician Coriolis and he did a lot of work on understanding what goes on in rotating reference frames and the other force represented by the matrix C are centripetal torques. 

Once again we have our two link robot represented symbolically and this time we're going to investigate the Coriolis and centripetal torques. And that's done using the method called coriolis and the arguments to that are the joint angles, there they are and the joint velocities; Q1D and Q2D.

It takes a moment to compute and this is the Coriolis and centripetal matrix. It's a 2 x 2 matrix and we can see, once again that it comprises a number of kinematic and dynamic parameters of a robot and we can also see that it's a function of the configuration of a robot. Interestingly, in this case it’s only a function of Q2. 

For a simple 2 link robot manipulator, the C matrix looks something like this. We can see that the elements are a function of the joint angle, in this case just a single joint angle. Joint 2 angle Q2 and also the velocities of the two joints Q1 dot and Q2 dot.

Now the torque, due to the Coriolis and centripetal effects can be derived by multiplying the matrix C by the vector of joint velocities and if we expand that out, we have the expression for the Coriolis and centripetal talks acting on Joint 1 and acting on Joint 2. 

Let's consider the case where Joint 1 is not moving that’s Q1 dot is equal to zero but let's say that Joint 2 is moving so Q2 dot is not equal to zero. Then although this term here is equal to zero. This term here will be finite, that is the motion of Joint 2 at constant velocity is going to induce a torque on Joint 1 and that's again something that the Joint 1 motion controller is going to have to fight against, going to have to oppose in order to keep Joint 1 stationary.

And similarly we can see that for Joint 2 constant velocity motion of Joint 1 is going to induce a Coriolis and centripetal effect on Joint 2. Now if we look at these terms, we can see that some of them are a function of a particular joint velocity squared and these are the centripetal talks and some of them are a product of one joint velocity multiplied by a different joint velocity and these are the Coriolis terms. So this matrix C has got some terms which are the products of velocities of different joints and some are joint velocity squared, Coriolis and centripetal effects.