1. ASME DETC 1998 1 Robot Manipulators and Singularities Vijay Kumar

2. ASME DETC 1998 2 Outline l Jacobian matrix for a serial chain manipulator l Singularities l Parallel manipulator

3. ASME DETC 1998 3 Serial Chain Linkages 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). Axis 1 Axis 2 y z O u2u2 u1u1 x Axis n

4. ASME DETC 1998 4 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). Axis 1 Axis 2 y z O u2u2 u1u1 x Axis n

5. ASME DETC 1998 5 Serial chain linkages Assume l Single degree-of-freedom, axial joints l ith joint twist  T i = S i  i u revolute joints: u prismatic joints: l Velocity equations T = T 1 + T 2 + … + T n “Standard form”

6. ASME DETC 1998 6 Serial chain equations Jacobian matrix l Geometric significance of the columns of the matrix l Matrix can be constructed by inspection l Physical insight into the kinematic performance End effector twist Joint ratesJacobian matrix Axis 1 Axis 2 y z O u2u2 u1u1 x Axis n

7. ASME DETC 1998 7 Jacobian matrix Axis 1 Axis 2 Axis 3 Axis 4 Axis 5 Axis 6

8. ASME DETC 1998 8 Singularities l C 3 = 0 l S 5 = 0 l Jacobian matrix Axis 1 Axis 2 Axis 3 Axis 4 Axis 5 Axis 6 ml n z y Axis 1 Axis 2 Axis 3 Axis 4 Axis 5 Axis 6

9. ASME DETC 1998 9 Example Axis 1 Axis 2 Axis 3 Axis 4 Axis 5 Axis 6 Singularities l C 3 = 0 l S 5 = 0 l 1 2 3 4 5 6

10. ASME DETC 1998 10 Example Revolute Joints Prismatic Joints 4 1 2 3 5 6 1 2 3 5 6 x z

11. ASME DETC 1998 11 Singularities l Algebra Jacobian matrix becomes singular l Geometry The joint screws (lines) are linearly dependent l Kinematics The manipulator (instantaneously) loses one or more degrees of freedom l Statics There exists one or more wrenches that can be resisted without turning on the actuators

12. ASME DETC 1998 12 Case 1 C 3 = 0 u Zero pitch wrench reciprocal to all joint screws u Line intersects all six joint axes u Rows 1, 5, and 6 are dependent u It is not possible to effect the twist [n l S 2, 0, 0, 0, - l S 2, n S 2 +mC 2 ] T Singularities Axis 1 Axis 2 Axis 3 Axis 4 Axis 5 Axis 6 ml z y

13. ASME DETC 1998 13 Case 2 S 5 = 0 u Axes 4 and 6 are dependent u Joints 4 and 6 have the same instantaneous motions u The end effector loses a degree of freedom Singularities (continued) Axes 4 and 6 become colinear Axis 4 Axis 5 Axis 6 P Q Link 3

14. ASME DETC 1998 14 Case 3 u Point of concurrence of axes 4, 5, and 6 lies on the plane defined by axes 1 and 2 u Zero pitch wrench reciprocal to all the joint screws u Line intersects or is parallel to all joint axes u Rows 1 and 5 are dependent u The end effector cannot move along the twist: [-n, 0, 0, 0, 1, 0] T Singularities (continued) Axis 1 Axis 2 Axis 3 Axis 4 Axis 5 Axis 6 ml

15. ASME DETC 1998 15 Singularities: More Examples P a2a2 a3a3 Axes 4 and 6 become colinearManipulator is completely flexed/extended Axis 4 Axis 5 Axis 6 P Q Spherical wrist Link 3

16. ASME DETC 1998 16 Singularities: More Examples l Case 1: the manipulator is completely extended or flexed l Case 2: the tool reference point lies on axis 1 l Case 3: orientation singularity Axes 4 and 6 are colinear P a2a2 a3a3

17. ASME DETC 1998 17 Singular Structure Six degree of freedom robot manipulator with an anthropomorphic shoulder and wrist Three axes intersecting at a point

18. ASME DETC 1998 18 Special Third Order System: Type 2 l System consists of zero pitch screws on all lines through a point l There are no members with other pitches l Screw system of spherical joint l Self-reciprocal

19. ASME DETC 1998 19 Manipulator Screw System

20. ASME DETC 1998 20 Parallel Manipulators Stewart Platform l Each leg has five passive joints and one active (prismatic joint) l There is a zero pitch wrench reciprocal to all five passive joints. Call it S i for Leg i. l The net effect of the prismatic joint must be to produce this zero pitch wrench. u Twists of freedom is a fifth order screw system defined by the five passive joints u Constraint wrench system is defined by the zero pitch reciprocal screw l The end effector wrench is the sum of the wrenches exerted by the six actuators (acting in parallel) Axis of the reciprocal wrench

21. ASME DETC 1998 21 Parallel Manipulators l The columns of the transpose of the Jacobian matrix are the coordinates of the reciprocal screws. l The equations for force equilibrium (statics) for parallel manipulators are “isomorphic” to the equations for rate kinematics for serial manipulators. l A parallel manipulator is singular when u Any of its serial chains becomes singular (kinematic singularity) u The set of reciprocal screws (S i ) becomes linearly dependent

22. ASME DETC 1998 22 Parallel Manipulators: Example l Each serial chain consists of two revolute joints and 1 prismatic joint. l 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. l Actuator i produces a pure force along the screw S i l The manipulator is singular when the axes of the reciprocal screws intersect at a point (or become parallel) l At this singularity, the actuators cannot resist a moment about the point of intersection (or a force perpendicular to the all the three axes) 1 2 3 S3S3 S2S2 S1S1