Kinematic Modeling

Kinematic Modeling

Direct

Inverse

Analytical Jacobian

Geometric Jacobian

Interplay between the feasible twists of and the constraint wrenches acting on the various rigid bodies of a mechanical system, when the system moves under static, conservative conditions.

We derive first the relation between the twist of the robot EE and the set of joint rates, which is given by a linear transformation induced by the robot Jacobian matrix.

Decoupled and planar architectures are treated as special cases of six-joint robots.

Used for

Velocity Analisis of Serial Manipulators

If the end-effector pose is specified in terms of a minimal number of parameters in the operational space If the end-effector pose is specified in terms of a minimal number of parameters in the operational space

The translational velocity of the end-effector frame can be expressed as the time derivative of vector pe, representing the origin of the end-effector frame with respect to the base frame

For what concerns the rotational velocity of the end-effector frame, the minimal representation of orientation in terms of three variables φe can be considered.

Its time derivative ˙ φe in general differs from the angular velocity vector defined above. In any case, once the function φe(q) is known, it is formally correct to consider the Jacobian obtained as

Computing the Jacobian Jφ(q) as∂φe/∂q is not straightforward, since the function φe(q) is not usually available in direct form, but requires computation of the elements of the relative rotation matrix.

Upon these premises, the differential kinematics equation can be obtained as the time derivative of the direct kinematics equation

where the analytical Jacobian

the equation relating the angular velocity ωe to the time derivative of the Euler angles ˙ φe

The goal of the differential kinematics is to find the relationship between the joint velocities and the end-effector linear and angular velocities.

it is desired to express the end-effector linear velocity ˙ pe and angular velocity ωe as a function of the joint velocities ˙ q.

Represents the manipulator differential kinematics equation. The (6×n) matrix J is the manipulator geometric Jacobian

Antisymetric Matrix

The manipulator direct kinematics equation in (3.1) describes the end-effector pose, as a function of the joint variables, in terms of a position vector and a rotation matrix.

which, differentiated with respect to time, gives the identity

The matrix S(t) is so that its symmetric elements with respect to the main diagonal represent the components of the vector ω(t)=[ωx ωy ωz ]T in the form

Velocity Propagation

Using the DH matrix

Obtain the Jg columns that relate the gripper speeds

the vectors zi−1, pe and pi−1 are all functions of the joint variables

First zi−1 is given by the third column of the rotation matrix R0 i−1 where z0 = [0 0 1] T allows the selection of the third column

pe is given by the first three elements of the fourth column of the transformation matrix T0e

Then we get Jg where

Relationship between Cartesian position and joint values.

Knowing the Cartesian position of the terminal element, joint values can be determined.

If X,Y

Belongs to workspace

Doesn´t belongs to workspace

No solution

Single solution, multiple solutions, infinite solutions

Methods

Algebraic Geometric

From the homogeneous transformation matrix

Cinematic decoupling

Isis Danae Sánchez Hernández --- 1620197

[1]B. Siciliano, L. Sciavicco, L. Villani and G. Oriolo, Robotics. Modelling, Planning and Control. London: Springer-Verlag London, 2010.