Kinematics - prisma.unina.it · Prof.Fanny Ficuciello Robotics for Bioengineering • Kinematics...

Prof. Fanny Ficuciello Robotics for Bioengineering • Kinematics

End-effector position and orientation (pose) are expressed as a function of the joint variables of the mechanical structure with respect to a reference frame Rotation matrix Representation of orientation (Euler angles, angle and axis, unit quaternion) Homogeneous transformations Direct kinematics Open chain Denavit–Hartenberg convention Kinematics of typical manipulator structures ... … …

Kinematics


A manipulator consists of a series of rigid bodies (links) connected by means of kinematic pairs or joints.

Kinematic chain open (only one sequence of links) closed (a sequence of links forms a loop)

Degrees of freedom (DOFs) are typically associated with joint articulations and constitute the joint variables

Direct Kinematics


Direct kinematic function

Base Frame and End-effector Frame


Two-link Planar Arm


Open Chain


Choose axis zi along the axis of Joint i + 1 Locate the origin Oi at the intersection of axis zi with the common normal to axes zi−1

and zi. Also, locate Oi at the intersection of the common normal with axis zi−1

Choose axis xi along the common normal to axes zi−1 and zi with direction from Joint i to Joint i + 1

Choose axis yi so as to complete a right-handed frame

Denavit–Hartenberg Convention


The Denavit–Hartenberg convention gives a nonunique definition of the link frame in the following cases:

For Frame 0, only the direction of axis z0 is specified; then O0 and x0 can be arbitrarily chosen

For Frame n, since there is no Joint n+1, zn is not uniquely defined while xn has to be normal to axis zn−1. Typically, Joint n is revolute, and thus zn is to be aligned with the direction of zn−1

The common normal between two lines is the line containing the minimum distance segment between the two lines

When two consecutive axes are parallel, the common normal between them is not uniquely defined

When two consecutive axes intersect, the direction of xi is arbitrary

When Joint i is prismatic, the direction of zi−1 is arbitrary

Denavit–Hartenberg Convention


distance between Oi and Oi

coordinate of Oi along zi−1

angle between axes zi−1 and zi about axis xi to be taken positive when rotation is made counter-clockwise

angle between axes xi−1 and xi about axis zi−1 to be taken positive when rotation is made counter-clockwise

and are always constant if Joint i is revolute the variable is ϑi

if Joint i is prismatic the variable is di

Denavit–Hartenberg Parameters


Coordinate Transformation


1. Find and number consecutively the joint axes; set the directions of axes z0 . . zn−1

2. Choose Frame 0 by locating the origin on axis z0; axes x0 and y0 are chosen so as to obtain a right-handed frame

Execute steps from 3 to 5 for i = 1, . . . , n − 1:3. Locate the origin Oi at the intersection of zi with the common normal to axes zi−1

and zi. If axes zi−1 and zi are parallel and Joint i is revolute, then locate Oi so that di = 0; if Joint i is prismatic, locate Oi at a reference position for the joint range, e.g., a mechanical limit

4. Choose axis xi along the common normal to axes zi−1 and zi with direction from Joint i to Joint i + 1

5. Choose axis yi so as to obtain a right-handed frame

Composition of Rotation Matrices


To complete:6. Choose Frame n; if Joint n is revolute, then align zn with zn−1, otherwise, if Joint n

is prismatic, then choose zn arbitrarily. Axis xn is set according to step 47. For i = 1, . . . , n, form the table of parameters8. On the basis of the parameters in 7, compute the homogeneous transformation

matrices for i = 1, . . . , n9. Compute the homogeneous transformation

… and finally

Composition of Rotation Matrices


Three-link Planar Arm


Spherical Arm


Anthropomorphic Arm


Spherical Wrist


Stanford Manipulator


Anthropomorphic Arm with Spherical Wrist


DLR Manipulator


configuration where the last joint is so that

a connecting device between the end-effector of the anthropomorphic torso and the base frames of the two manipulators permits keeping the ‘chest’ of the humanoid manipulator always orthogonal to the ground

Humanoid Manipulator

Kinematics - prisma.unina.it · Prof.Fanny Ficuciello Robotics for Bioengineering • Kinematics...

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