## The Analytic Jacobian

In Robotics | No commentNow we introduce a variant of the Jacobian matrix that can relate our angular velocity vector back to our rates of change of the roll, pitch and yaw angles.

Now we introduce a variant of the Jacobian matrix that can relate our angular velocity vector back to our rates of change of the roll, pitch and yaw angles.

We previously learnt how to derive a Jacobian which relates the velocity of a point, defined relative to one coordinate frame, to the velocity relative to a different coordinate frame. Now we extend that to the 3D case.

The Jacobian matrix provides powerful diagnostics about how well the robot’s configuration is suited to the task. Wrist singularities can be easily detected and the concept of a velocity ellipse is extended to a 3-dimensional velocity ellipsoid.

As we did for the simple planar robots we can invert the Jacobian and perform resolved-rate motion control.

We resume our analysis of the 6-link robot Jacobian and focus on the rotational velocity part.

We have a quick revision of the skew-symmetric matrix. If you’re comfortable with this topic then go straight on to the next section.

For a real 6-link robot our previous approach to computing the Jacobian becomes unwieldy so we will instead compute a numerical approximation to the forward kinematic function.

A body moving in 3D space has a translational velocity and a rotational velocity. The combination is called spatial velocity and is described by a 6-element vector.

We will learn about the relationship, in 3D, between the velocity of the joints and the velocity of the end-effector — the velocity kinematics. This relationship is described by a Jacobian matrix which also provides information about how easily the end-effector can move in different Cartesian directions. To do this in 3D we need to learn about rate of change of orientation and the concept of angular velocity.

This video gives summary of velocity Kinematics in 2D

We can also derive a Jacobian which relates the velocity of a point, defined relative to one coordinate frame, to the velocity relative to a different coordinate frame.

We extend what we have learnt to a 3-link planar robot where we can also consider the rotational velocity of the end-effector.

We will introduce resolved-rate motion control which is a classical Jacobian-based scheme for moving the end-effector at a specified velocity without having to compute inverse kinematics.

The end-effector is not able to move equally fast in all directions, and that in fact depends on the pose of the robot. We will introduce the velocity ellipse to illustrate this.

By inverting the Jacobian matrix we can find the joint velocities required to achieve a particular end-effector velocity, so long as the Jacobian is not singular.

For a simple 2-link planar robot we introduce and derive its Jacobian matrix, and also introduce the concept of spatial velocity.

We introduce the relationship between the velocity of the robot’s joints and the velocity of the end-effector in 3D space.

We will learn about the relationship, in 2D, between the velocity of the joints and the velocity of the end-effector — the velocity kinematics. This relationship is described by a Jacobian matrix which also provides information about how easily the end-effector can move in different Cartesian directions.

This video gives summary of inverse kinematics.

An alternative for smooth motion between poses is Cartesian interpolated motion which leads to straight line motion in 3D space.