D R . T A R E K A . T U T U N J I

M E C H A T R O N I C S E N G I N E E R I N G D E P A R T M E N T

P H I L A D E L P H I A U N I V E R S I T Y , J O R D A N

2018

Robotics: Differential Kinematics

Outline

Geometric Jacobian

Jacobian of Manipulator Structures

Analytical Jacobian

Kinematic Singularities

Inverse Kinematics Algorithms

Differential Kinematics

Differential kinematics gives the relationship between the joint variables and the end-effector

linear and angular velocities

The Jacobian Matrix

Defines the relationship between joint and workspace velocities.

Defines relationship between forces/torques between spaces

Studies the singular configurations

Defines numerical procedures for solving Inverse Kinematics problem

Studies the manipulability properties

Jacobian: Geometric and Analytical

The Jacobian calculates the linear and angular velocities of the robot joints. In particular, the interest is in the end-effector.

There are two formats which might lead to different results for the part related to the rotational velocity.

Geometric Jacobian The relationship between the joint velocities and corresponding end

effector linear and angular velocities

Analytical Jacobian The end-effector pose is expressed with reference to a minimal

representation in the operational space.

The Jacobian is computed via differentiation of the direct kinematics with respect to the joint angles.

Geometric Jacobian

The general direct kinematics equation for n-DOF manipulator can be given as

Geometric Jacobian

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

J is the manipulator geometric Jacobian

J is 6xn matrix

Jacobian Computation

The Jacobian can be partitioned into 3x1 column vectors Jpi and Joi

Jacobian of Three-link Planar Arm

Jacobian of Three-link Planar Arm

Jacobian of Three-link Planar Arm

Jacobian of Anthropomorphic Arm

Jacobian of Anthropomorphic Arm

Jacobian of Anthropomorphic Arm

Analytical Jacobian

Compute the Jacobian via differentiation of the direct kinematics function w.r.t. the joint variables.

Analytical Jacobian: Example

Three-Planar Arm Class Notes

Rotational Velocity vs. Angular Velocity

Jacobian Demonstration

https://www.youtube.com/watch?v=qevzNE_hL_k

Kinematic Singularities

Kinematic singularities are of interest for the following reasons:

Mobility of the structure is reduced

Infinite solutions to the inverse kinematics may exist

Small velocities in the operational space may cause large velocities in the joint space

Singularities can be classified into:

Boundary singularities

Internal singularities

Kinematic Singularities

The determinant vanishes at

Spherical Wrist at Singularity

Anthropomorphic arm at singularity

Elbow Singularity Shoulder Singularity

https://www.youtube.com/watch?v=6Wmw4lUHlX8

Inverse Differential Kinematics

Inverse Kinematics has closed-form solutions only for manipulators with simple structures.

Limitations are related to the highly nonlinear equations.

Differential kinematics equations represent a linear mapping between the joint velocity space and the operational velocity space ??

Inverse Differential Algorithms

Open-loop Inverse Jacobian

Inverse Differential Algorithms

Closed-loop Inverse Jacobian

Inverse Differential Algorithms

Jacobian Transpose

Inverse Differential Algorithms

Second-order Inverse Jacobian

Comparisons among Inverse Kinematic Algorithms

Consider the 3-link planar arm with a1=a2=a3=0.5m

Initial posture: q = [ p -p/2 -p/2] rad

Corresponding to p = [ 0 0.5] and f= 0 rad A circular path of radius 0.25m and center at (0.25,0.5) m is assigned for the end-effector. The motion is set to

Algorithm: Open loop Inverse Jacobian

Algorithm: Closed-loop Inverse Jacobian

Algorithm: Jacobian Transpose

Reference

Siciliano, Sciavicco, Villani, and Oriolo. Robotics: Modeling, Planning, and Control. Advanced Textbooks in Control and Signal Processing. Springer 2009