Kinematics of a Redundant Anthropomorphic Arm with Seven Degrees of Freedom

model

line

profile

frame

number

0

1

2

3

4

5

6

7

kinematics

forward

inverse

end effect

constrain space

position (Cartesian xyz)

x-y

z

orientation (Euler angles zyz)

α

β

γ

The anthropomorphic arm with seven degrees of freedom (DoF) includes redundancy, since the number of joints (

n=7

) is greater than the dimension of the manipulation variables, which have three position coordinates and three orientation angles (

m=6

). Although the set of nonlinear trigonometric equations cannot typically be solved analytically, there are some robot structures that are solvable analytically. Pieper proposed a decoupling method [1]; the sufficient condition for solvability is when the 6-DoF robot has three consecutive revolute joints with axes intersecting in one point. Another generalized inverse is the inertia-weighted pseudoinverse Jacobian matrix, a kind of resolved motion rate control technique, proposed by Whitney [2], which is used to minimize energy by using the inertia matrix as the weighting matrix.

This Demonstration solves the inverse kinematics of a 7-DoF anthropomorphic arm with a decoupling hybrid algorithm. It uses a Newton–Raphson numerical technique and a pseudoinverse Jacobian matrix to solve the shoulder and elbow 4-DoF component for end-effector position and applies an analytical method to solve the wrist 3-DoF component for end-effector orientation.