Just a quick reminder with another example Recap D-H Just a quick reminder with another example

6 DOF revolute robot

6 DOF revolute robot

6 DOF revolute robot

Inverse Kinematic solution As previously, to solve for the 6 angles, successively premultiply both matrices with An-1 matrices, starting with A1-1 . At each stage it is possible to yield useful results… …from large matrices and complex trig simplification due to many coupled angles. Hence, results summarised here. If you really want to know read Niku p87-93 (76-80) (who obtained it from Paul, R.P. “Robot Manipulators, Mathematics, Programming & Control” MIT Press, 1981)

Summary for 6 d.o.f revolute robot Don’t panic this is all in the Kinematics Data sheet!

Jacobian & Differential Motions Look at velocity relationships rather than position.

Robot Dynamics Previously we have only considered static robot mechanisms but they move! Need to consider the velocity relationships between different parts of the robot mechanism. Like the engineering dynamics you have already studied, we consider small movements i.e. a differential motion which is over a small period of time.

Consider a 2 DOF Mechanism VB/A VB (if VA = 0) VA VB = VA + VB/A

2 DOF mechanism This equation can be expanded and written in matrix form: Reason why this differs in second line is because you need to take into account that VA/B coordinate system changes due to point A moving i.e. axes x-z at point A are moving hence VB = VA + ω x rB/A + (VB/A)xyz (see Hibbeler dynamics p356) where middle bit is the angular velocity effect caused by rotation of the second link relative to the reference frame.

2 DOF mechanism Or differentiate the equations that describe the location of point B:

If you didn’t understand the last slide… All dx, dy and dθ should be: Hence:

Jacobian It is a representation of the geometry of the elements of a mechanism in time. In robotics, it relates joint velocities to Cartesian velocities of a point of interest e.g. the end-effector. End-effector differential motions Joint differential motions displacement motions rotation motions

Jacobian dx, dy, dz (in [D]) represent the differential motions of the end-effector along the x-, y-, z-axes respectively. δx, δy, δz (in [D]) represent the differential rotations of the end-effector along the x-, y-, z-axes respectively. End-effector differential motions Joint differential motions displacement motions rotation motions

Jacobian [Dθ] represents the differential motions of the joints If matrices [D] and [Dθ] are divided by dt, they will represent the velocities. End-effector differential motions Joint differential motions displacement motions rotation motions

Jacobian Matrix compared with kinematics and with an alternative notation Forward Jacobian Matrix Kinematics Inverse Jacobian Matrix: Relationship between joint space velocity with world space velocity Joint Space World Space

Jacobian Jacobian is a function of q, a joint variable (e.g. length or angle), it is not a constant!

Example 1 J, the Jacobian of a robot at a particular time is given. Calculate the linear and angular differential motions of the robots “Hand” frame for the given joint differential motions Dθ

Example 1

Differential Motions of a Frame Differential motions of a frame – relate to differential motions of the robot

Differential translation Trans (dx, dy, dz)

Differential Rotations Approximate: sin δx ≈ δx (in radians) cos δx ≈ 1

Differential Rotation @ some random Axis Any order SAME ANSWER velocities “commutative” Neglect higher order differentials, e.g. δxδy→0

Example 2 Find total differential transformation from 3 small rotations: δx = 0.1, δy = 0.05, δz = 0.02 radians.

Differential Transformations of a Frame If you have an original frame T Unit or Identity matrix I

Differential Transformations of a Frame [dT ] expresses the change in the frame after the differential transformation: Differential operator

Differential Transformations of a Frame Differential operator Δ This is not a transformation nor a frame, merely an operator.

Example 3 (part A)

Example 3A - solution

Differential Change – Example 3B Find the location & orientation of frame B

Differential changes between frames Differential operator Δ Is relative to a fixed reference frame, UΔ Define a differential operator TΔ relative to the current frame T. As TΔ relative to the current frame T, we need to post-multiply (as we did in lecture 1).

Differential changes between frames

After some long derivation: TΔ is made to look like Δ matrix where:

Example 3 Again Part C Find BΔ: From Boriginal matrix From the differential operator OR differential transformation description

Example 3C continued

Example 3 – part D, alternative Alternatively BΔ can be found using: Same result as before!

Differential Motion of robot From earlier slide: How are differential motions accomplished by the robot? End-effector differential motions Joint differential motions displacement motions rotation motions

To Calculate the Jacobian Next lecture!

Jacobian Matrix Inverse Jacobian Singularity Avoid it rank(J)<min{6,n}, Jacobian Matrix is less than full rank Jacobian is non-invertible Occurs when two or more of the axes of the robot form a straight line, i.e., collinear Avoid it