Post on 16-Dec-2015
Chapter 4: Inverse Kinematics
1
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSOutline:
• Introduction• Solvability • Manipulator subspace when n<6• Algebraic vs. Geometric• Example: Kinematics of PUMA Robot
Chapter 4: Inverse Kinematics
2
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSIntroduction:
• Direct kinematics:Given the joint angles calculate the position and
orientation of the tool {T} relative to the station {S}• Inverse kinematics:
Given the desired position and orientation of the tool {T} relative to the station {S}, how to calculate the set of joint angles that give the desired result? – Find {W} from {T}; Find {B} from {S};
– Solve for joint angles
WTTBST
BWT
Chapter 4: Inverse Kinematics
3
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSSolvability:
• Nonlinear problem: solve to find the values of
• Linear and nonlinear problems? • Ex.: Puma manipulator
01 1( , ,..., )B
W n nT T
1 1, ,..., n
11 12 13
21 22 230
31 32 33
0 0 0 1
x
yn
z
r r r P
r r r PT
r r r P
12 values find
12 equations and 6 unknowns, rotation part (r11 … r33), only 3 independent
Chapter 4: Inverse Kinematics
4
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSSolvability:
• Nonlinear problem: solve to find the values of
• Linear and nonlinear problems? • Ex.: Puma manipulator
01 1( , ,..., )B
W n nT T
1 1, ,..., n
11 12 13
21 22 230
31 32 33
0 0 0 1
x
yn
z
r r r P
r r r PT
r r r P
12 values find
12 equations and 6 unknowns, rotation part (r11 … r33), only 3 independent
Chapter 4: Inverse Kinematics
5
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSSolvability:
3 independent equations for the orientation and 3 independent equations for the position Nonlinear equations that are difficult to solve. – Note that these were for simple links α = 0, 90, -90, … and many
(d & a) = 0.– General case (α, d, & a) have other nonzero valuesMore complex case.
We must concern on – Existence of solution?– Multiple solutions?– Method of solution?
Chapter 4: Inverse Kinematics
6
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSSolvability:
• Existence of solutionExistence or nonexistence of a solution defines the workspace of the manipulator.
• Workspace (W.S.): (Figure)– The volume of the space that the E.E. of the robot can reach.– A solution to exist the desired position & orientation (goal)
must lie in the W.S.
• Two types of workspaces:– Dextrous W.S.: volume of the W.S. in which the E.E. can reach
with all the orientations.– Reachable W.S.: volume of the W.S. in which the E.E. can reach in
at least one orientation. Dextrous W.S. is a subset of the reachable W.S.
Chapter 4: Inverse Kinematics
8
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSSolvability:
• Example: if l1 = l2 determine the dextrous and reachable W.S.
Chapter 4: Inverse Kinematics
9
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSSolvability:
• Factors that affect the W.S. of a robot:– Limitations on joint anglesGenerally: min < θi < max. Most of the robots have limitations on their joint angles. The W.S. is reduced; this may affect the attained positions or the
number of orientations for the attained position. • Ex.: for the previous sketch 0 ≤ θi ≤ 180 sketch the W.S.
The same positions can be reached, however in only one orientation.
– Limitations on the number of DoF:A 3D-space goal is not attainable by a manipulator of DoF < 6.For the manipulator of the previous example can not attain a position of nonzero z-value.