Jacobians:

1. Jacobians: Velocities and Static Force Amirkabir University of TechnologyComputer Engineering & Information Technology Department http://ce.aut.ac.ir/~shiry/lecture/robotics-2004/robotics04.html

2. Differentiation of position vectors Derivative of a vector: We are calculating the derivative of Q relative to frame B.

3. Differentiation of position vectors A velocity vector may be described in terms of any frame: We may write it: Speed vector is a free vector Special case: Velocity of the origin of a frame relative to some understood universe reference frame

4. Example 5.1 Both vehicles are heeding in X direction of U 100 mph A fixed universal frame 30 mph

5. Angular velocity vector: Linear velocity  attribute of a point Angular velocity  attribute of a body Since we always attach a frame to a body we can consider angular velocity as describing rational motion of a frame.

6. Angular velocity vector: describes the rotation of frame {B} relative to {A} direction of indicates instantaneous axis of rotation Magnitude of indicates speed of rotation In the case which there is an understood reference frame:

7. Linear velocity of a rigid body We wish to describe motion of {B} relative to frame {A} If rotation is not changing with time:

8. Rotational velocity of a rigid body Two frames with coincident origins The orientation of B with respect to A is changing in time. Lets consider that vector Q is constant as viewed from B.

9. Rotational velocity of a rigid body Is perpendicular to and Magnitude of differential change is: Vector cross product

10. Rotational velocity of a rigid body In general case:

11. Simultaneous linear and rotational velocity We skip 5.4!

12. Motion of the Links of a Robot Written in frame i At any instant, each link of a robot in motion has some linear and angular velocity.

13. Velocity of a Link Remember that linear velocity is associated with a point and angular velocity is associated with a body. Thus: The velocity of a link means the linear velocity of the origin of the link frame and the rotational velocity of the link

14. Velocity Propagation From Link to Link • We can compute the velocities of each link in order starting from the base. • The velocity of link i+1 will be that of link i, plus whatever new velocity component added by joint i+1.

15. Rotational Velocity • Rotational velocities may be added when both w vectors are written with respect to the same frame. • Therefore the angular velocity of link i+1 is the same as that of link i plus a new component caused by rotational velocity at joint i+1.

16. Velocity Vectors of Neighboring Links

17. Velocity Propagation From Link to Link Note that: By premultiplying both sides of previous equation to:

18. Linear Velocity • The linear velocity of the origin of frame {i+1} is the same as that of the origin of frame {i} plus a new component caused by rotational velocity of link i.

19. Linear Velocity Simultaneous linear and rotational velocity: By premultiplying both sides of previous equation to:

20. Prismatic Joints Link For the case that joint i+1 is prismatic:

21. Velocity Propagation From Link to Link • Applying those previous equations successfully from link to link, we can compute the rotational and linear velocities of the last link.

22. A 2-link manipulator with rotational joints Example 5.3 Calculate the velocity of the tip of the arm as a function of joint rates?

23. Example 5.3 Frame assignments for the two link manipulator

24. Example 5.3 We compute link transformations:

25. Example 5.3 Link to link transformation

26. Example 5.3 Velocities with respect to non moving base

27. Derivative of a Vector Function • If we have a vector function r which represents a particle’s position as a function of time t:

28. Vector Derivatives • We’ve seen how to take a derivative of a vector vs. A scalar • What about the derivative of a vector vs. A vector?

29. Jacobian • A Jacobian is a vector derivative with respect to another vector • If we have f(x), the Jacobian is a matrix of partial derivatives- one partial derivative for each combination of components of the vectors • The Jacobian is usually written as j(f,x), but you can really just think of it as df/dx

30. Jacobian

31. Partial Derivatives • The use of the ∂ symbol instead of d for partial derivatives just implies that it is a single component in a vector derivative.

32. Jacobian Chain rule J(X)

33. Jacobian In the field of robotics, we generally speak of Jacobians which relate joint velocities to Cartesian velocities of the tip of the arm.

34. Jacobian For a 6 joint robot the Jacobian is 6x6, q. is a 6x1 and v is 6x1. The number of rows in Jacobian is equal to number of degrees of freedom in Cartesian space and the number of columns is equal to the number of joints.

35. Jacobian In example 5.3 we had: Thus: And also:

36. Jacobian • Jacobian might be found by directly differentiating the kinematic equations of the mechanism for linear velocity, however there is no 3x1 orientation vector whose derivative is rotational velocity. Thus we get Jacobian using successive application of:

37. Singularities Given a transformation relating joint velocity to Cartesian velocity then Is this matrix invertible? ( Is it non singular)

38. Singularities Singularities are categorized into two class: • Workspace boundary singularities: Occur when the manipulator is fully starched or folded back on itself. • Workspace interior singularities: Are away from workspace boundary and are caused by two or more joint axes lining up. All manipulators have singularity at boundaries of their workspace. In a singular configuration one or more degree of freedom is lost. ( movement is impossible )

39. Example 5.4 In example 5.3 we had: Workspace boundary singularities

40. Example 5.5 As the arm stretches out toward q2=0 both joint rates go to infinity

41. Static Forces in Manipulators Force and moments propagation To solve for joint torques in static equilibrium force exerted on link i by link i-1 torque exerted on link i by link i-1

42. Static Forces in Manipulators Solve for the joint torques which must be acting to keep the system in static equilibrium. Summing the force and setting them equal to zero Summing the torques about the origin of frame i

43. Static Forces in Manipulators Working down from last link to the base we formulate the force moment expressions Static force propagation from link to link: Important question: What torques are needed at the joint to balance reaction forces and moments acting on the links?

44. Work-energy Principle • The change in the kinetic energy of an object is equal to the net work done on the object.

45. Principle of Virtual Work External virtual work equals the internal virtual strain energy.

46. Jacobians in the Force Domain Work is the dot product of a vector force or torque and a vector displacement It can be written as: The definition of jacobian is So we have

47. Cartesian Transformation of Velocities and Static Forces General velocity of a body 3 x1 linear velocity 3 x1 angular velocity General force of a body 3 x1 force vector 3 x1 moment vector 6 x 6 transformations map these quantities from one frame to another.

48. Cartesian Transformation of Velocities and Static Forces (5.45) Since two frames are rigidly connected Where the cross product is the matrix operator

49. Cartesian Transformation of Velocities and Static Forces We use the term velocity transformation Description of velocity in terms of A when given the quantities in B

50. Cartesian Transformation of Velocities and Static Forces A force-moment transformation With similarity to Jacobians