#### Inverting the Jacobian Matrix

lesson

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.

lesson

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.

lesson

The relationship between world coordinates, image coordinates and camera spatial velocity has some interesting ramifications. Some very different camera motions cause identical motion of points in the image, and some camera motions leads to no change in the image at all in some parts of the image. Let’s explore at these phenomena and how we […]

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

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We learn the mathematical relationship between angular velocity of a body and the time derivative of the rotation matrix describing the orientation of that body.

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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.

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The image Jacobian depends not only on the image plane coordinates but also the distance from the camera to the points of interest. If this distance is not known, what can we do? Let’s look at how we can determine this distance, and how the optical flow equation can be rearranged to convert from observed […]

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We describe the velocity coupling terms of the robot as a matrix which represents how the torque on one joint depends on the velocity of other joints.

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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 […]

lesson

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.

lesson

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.