This video introduces the 6×6 adjoint representation of a 4×4 SE(3) transformation matrix and shows how it can be used to change the frame of reference of a twist or a screw. The 4×4 se(3) matrix representation of a 6-vector twist is also introduced. This se(3) representation is used in the matrix exponential in the

Any rigid-body transformation can be achieved from any other by following some 6-vector twist for unit time. The six coordinates of this twist are called the exponential coordinates. This video shows how the rigid-body transformation can be calculated using a matrix exponential with the se(3) matrix representation of the exponential coordinates. The matrix exponential maps

This video introduces the wrench 6-vector representation of forces and moments in three-dimensions, and it shows how to change the frame of representation of a wrench.

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This video introduces the product-of-exponentials formula, expressed in the space frame, for forward kinematics of an open-chain robot.

This video introduces the product-of-exponentials formula, expressed in the end-effector frame, for forward kinematics of an open-chain robot.

This video demonstrates the application of product-of-exponentials forward kinematics to an RRRP robot arm.

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This video introduces the Jacobian of a robot, and how it is used to relate joint velocities to end-effector velocities and endpoint forces to joint forces and torques. The notions of singularities, manipulability, the manipulability ellipsoid, and the force ellipsoid are also introduced.

This video introduces the space Jacobian, the Jacobian relating joint velocities to the end-effector twist expressed in the space frame.

This video introduces the body Jacobian, the Jacobian relating joint velocities to the end-effector twist expressed in the body frame (a frame at the end-effector).

This video derives the relationship between the end-effector wrench and joint forces and torques using the Jacobian.

This video discusses robot singularities and Jacobians where the number of joints is not equal to the number of components of the end-effector twist or velocity, resulting in “tall” (“kinematically deficient”) and “fat” Jacobians.

This video describes the visualization of a robot’s ease of moving in different directions in terms of a manipulability ellipsoid, as well as scalar measures of the manipulability. Analogous ideas are presented describing the robot’s ability to apply wrenches as its end-effector.

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This video introduces the inverse kinematics problem–finding a set of joint positions that yield a desired end-effector configuration–as well as two ways of solving the problem: analytically and by an iterative numerical method.

This video introduces the Newton-Raphson root-finding method for numerical inverse kinematics. The end-effector configuration is represented by a minimum set of coordinates. Representation of the end-effector configuration as a transformation matrix is covered in the next video.

This video applies the Newton-Raphson root-finding method for numerical inverse kinematics when the end-effector configuration is represented as a transformation matrix.

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This video takes an example-based approach to the kinematics of closed chains, particularly parallel robots, including forward kinematics, inverse kinematics, inverse velocity kinematics, and statics.