Background Twists in Kinematics and Wrenches in Statics

1. BackgroundTwists in Kinematics and Wrenches in Statics Vijay Kumar

2. Rigid Body Motion P(t) = R(t) p + d(t) R 3´3 rotation matrix d 3´1 translation vector Alternatively, The motion of a rigid body is described by the matrix function of time Background

3. Rigid Body Motion P(t) = R(t) p + d(t) W(t) angular velocity matrix vO (t) velocity of a point on the rigid body coincident with the origin Rigid Body Velocity • The velocity of the rigid body is described by W(t) and vO (t).

4. Twist A 6´1 velocity vector: Or, a 4´4 velocity matrix: The twist of the rigid body contains all information about the instantaneous velocity of the rigid body. Velocity and Twist • The velocity of any point on the rigid body is simply:

5. There is a unique (screw) axis and a pitch associated with any twist vector. Let u be a unit vector so that: w= wu Decompose vO into components parallel and perpendicular to u: vO = vparallel + vperpendicular Let vparallel = hu, and define a vector rso that: vperpendicular = r´ w Any twist vector can be written as: (r, u) define the axis h is the pitch of the axis w is the intensity of the twist about the ISA. Instantaneous Screw Axis (ISA) u r O

6. The Instantaneous screw axis (ISA) is described by: u, the orientation of the axis r, the position vector to a point on the axis h, the pitch of the axis The rigid body velocity can be thought of as a twist about a screw axis with a certain amplitude. w - amplitude of the twist - the vector representing the axis The twist vector is written as: ISA, continued u r O

7. Velocity field associated with a rigid body • Instantaneous screw axis and the helicoidal vector field w ´ q vP Helix of pitch h and axis S passing through the point P P hw q O The velocity field or the set of all velocity vectors forms a helicoidal vector field. Let the origin be on the instantaneous screw axis corresponding to the twist. The velocity of any point P consists of a component parallel to the instantaneous screw axis and another velocity component normal to it. The velocity component normal to the screw axis is also normal to the normal from the point to the screw axis and is proportional both to the length of that normal and to the angular velocity, w. It can be represented by the expression w´q, where q is the normal vector from the screw axis to the point.

8. Revolute joint Screws of zero and infinite pitch • Prismatic joint u u z z Axis O O q y y d Axis u = [0, 0, 1]T, r = [a, b, c]T, r ´ u = [b, -a, 0]T u = [0, 0, 1]T • The motion of the rigid body due to a revolute joint can be described as a twist about a screw of zero pitch. • The motion of the rigid body due to a prismatic joint can be described as a twist about a screw of infinite pitch.

9. What is a Wrench? Any system of forces and couples can be reduced to a pure force along an axis l, and a pure couple parallel to l. This combination of a force and a couple is called a wrench.

10. Wrenches • Resultant force and couple • Equipollent to a force F along l and a couple C parallel to l

11. The screw axis is described by: u, the orientation of the axis r, the position vector to a point on the axis h, the pitch of the axis The system of forces and couples can be thought of as a wrench about a screw axis with a certain intensity. F - intensity of the wrench - the vector representing the axis Any system of forces and couples can be reduced to a wrench vector which can be written as: u r O Screw Axis for a Wrench

12. Lines and Plucker Line Coordinates • Any line can be represented by 6 coordinates • (L, M, N, P, Q, R) • where • (L, M, N) T = v; • (P, Q, R) T = (r´v). • Note there are only four independent coordinates: • LP + MQ + NR = 0 • (L, M, N, P, Q, R) º (kL, kM, kN, kP, kQ, kR) A line can be written in the form where h=0.

13. Screws and Lines • An instantaneous screw axis or simply a screw is a line with a pitch associated with it. • A zero pitch screw can be identified with a line. • An infinite pitch screw can be identified with a line located at infinity.

14. A body subject to a pure force along the axis l A pure force is a wrench of zero pitch We can represent the force by a line. A body instantaneously undergoing a pure rotation about the axis l A pure rotation is a twist of zero pitch We can represent the rotation by a line Lines in Kinematics and Statics

15. A body subject to a pure couple A pure couple is a wrench of infinite pitch We can represent the couple by a line at infinity A body instantaneously undergoing a pure translation A pure translation is a twist of infinite pitch We can represent the translation by a line at infinity Lines in Kinematics and Statics (Cont’d)

16. Example P q2 a2 • Two revolute joints in series • Screw axis for Joint 1 u1 a1 z q1 O Axis 2 u2 y Axis 1 • Screw axis for Joint 2

17. Background: Open Chain Linkages Axis 2 u2 • System of screws: Open chain linkages • 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

18. Axis 2 u2 u1 Axis n z Axis 1 O y x Background: Closed Chain Linkages • System of screws: Open chain linkages • 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).

19. Background: Closed chain linkages Axis 2 u2 • System of screws for a closed chain • Consider a closed loop linkage with n joints. The twist of Link n-1 is given by: • If the twist of joint n is Tn, • In other words, u1 Link 1 Axis n-1 Link n-1 z Axis 1 O Axis n y x Link 0 Link 0 • The sum of the twists for all joints in a closed chain equals zero. • The joint twist in a closed chain is obtained from the other joint twists by simply adding (or subtracting) them.