Robot Manipulators and Singularities

1. Robot Manipulators and Singularities Vijay Kumar

2. Outline • Jacobian matrix for a serial chain manipulator • Singularities • Parallel manipulator

3. Serial Chain Linkages Axis 2 u2 • Velocity Equations • Let the end effector twist be T. • Consider two joints, 1 & 2. • The effect of twists about two joints connected in series is to produce a composite twist that is obtained by adding the two twists (in the same coordinate system). u1 Axis n z Axis 1 O y x

4. Axis 2 u2 u1 Axis n z Axis 1 O y x Serial Chain Linkages • Velocity Equations for a n-joint serial chain • The effect of twists about n joints connected in series is to produce a composite twist that is obtained by adding the n joint twists (in the same coordinate system).

5. Serial chain linkages • Assume • Single degree-of-freedom, axial joints • ith joint twist • Ti = Siai • revolute joints: • prismatic joints: • Velocity equations T = T1 + T2 + … + Tn “Standard form”

6. Axis 2 u2 u1 Axis n z Axis 1 O y x Serial chain equations • Jacobian matrix • Geometric significance of the columns of the matrix • Matrix can be constructed by inspection • Physical insight into the kinematic performance End effector twist Jacobian matrix Joint rates

7. Jacobian matrix Axis 5 Axis 6 Axis 2 Axis 3 Axis 4 Axis 1

8. Axis 1 Axis 2 Axis 3 Axis 4 Axis 5 Axis 6 Jacobian matrix • Singularities • C3= 0 • S5= 0 l m Axis 5 n Axis 6 y z Axis 2 Axis 3 Axis 1 Axis 4

9. Example 4 3 5 6 Axis 1 Axis 2 Axis 3 Axis 4 Axis 5 Axis 6 2 1 • Singularities • C3= 0 • S5= 0

10. 1 3 5 2 z x 6 Example 3 1 • Revolute Joints • Prismatic Joints 5 2 4 6

11. Singularities • Algebra Jacobian matrix becomes singular • Geometry The joint screws (lines) are linearly dependent • Kinematics The manipulator (instantaneously) loses one or more degrees of freedom • Statics There exists one or more wrenches that can be resisted without turning on the actuators

12. Singularities Case 1 C3= 0 • Zero pitch wrench reciprocal to all joint screws • Line intersects all six joint axes • Rows 1, 5, and 6 are dependent • It is not possible to effect the twist [n l S2, 0, 0, 0, - l S2, n S2+mC2]T l m Axis 5 z y Axis 2 Axis 3 Axis 6 Axis 4 Axis 1

13. Singularities (continued) Case 2 S5= 0 • Axes 4 and 6 are dependent • Joints 4 and 6 have the same instantaneous motions • The end effector loses a degree of freedom Axis 4 Link 3 Q Axis 5 Axis 6 P Axes 4 and 6 become colinear

14. l m Axis 2 Axis 5 Axis 3 Axis 6 Axis 4 Axis 1 Singularities (continued) Case 3 • Point of concurrence of axes 4, 5, and 6 lies on the plane defined by axes 1 and 2 • Zero pitch wrench reciprocal to all the joint screws • Line intersects or is parallel to all joint axes • Rows 1 and 5 are dependent • The end effector cannot move along the twist: [-n, 0, 0, 0, 1, 0]T

15. P a3 a2 Singularities: More Examples Axis 4 Link 3 Spherical wrist Q Axis 5 Axis 6 P Manipulator is completely flexed/extended Axes 4 and 6 become colinear

16. Singularities: More Examples P • Case 1: the manipulator is completely extended or flexed • Case 2: the tool reference point lies on axis 1 • Case 3: orientation singularity Axes 4 and 6 are colinear a3 a2

17. Singular Structure • Six degree of freedom robot manipulator with an anthropomorphic shoulder and wrist Three axes intersecting at a point

18. Special Third Order System: Type 2 • System consists of zero pitch screws on all lines through a point • There are no members with other pitches • Screw system of spherical joint • Self-reciprocal

19. Manipulator Screw System

20. Axis of the reciprocal wrench Parallel Manipulators • Stewart Platform • Each leg has five passive joints and one active (prismatic joint) • There is a zero pitch wrench reciprocal to all five passive joints. Call it Si for Leg i. • The net effect of the prismatic joint must be to produce this zero pitch wrench. • Twists of freedom is a fifth order screw system defined by the five passive joints • Constraint wrench system is defined by the zero pitch reciprocal screw • The end effector wrench is the sum of the wrenches exerted by the six actuators (acting in parallel)

21. Parallel Manipulators • The columns of the transpose of the Jacobian matrix are the coordinates of the reciprocal screws. • The equations for force equilibrium (statics) for parallel manipulators are “isomorphic” to the equations for rate kinematics for serial manipulators. • A parallel manipulator is singular when • Any of its serial chains becomes singular (kinematic singularity) • The set of reciprocal screws (Si) becomes linearly dependent

22. Parallel Manipulators: Example • Each serial chain consists of two revolute joints and 1 prismatic joint. • In the special planar three system, the joint screw reciprocal to the two revolute joints is the zero pitch screw in the plane whose axis intersects the two revolute joints. • Actuator i produces a pure force along the screw Si • The manipulator is singular when the axes of the reciprocal screws intersect at a point (or become parallel) • At this singularity, the actuators cannot resist a moment about the point of intersection (or a force perpendicular to the all the three axes) S3 S2 3 2 S1 1