1. Velocities and Static Force
Jacobians:
Velocities and Static Force
Amirkabir University of Technology Computer Engineering & Information Technology Department

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}
Attach a coordinate system to any body and study the motion of frames relative to one another.
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. Example 5.3
Calculate the velocity of the tip of the arm as a function of joint rates?
A 2-link manipulator with rotational joints

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

51. Example 5.8
Frames of interest with a force sensor

52. Next Course: Manipulator Dynamics
Amirkabir University of Technology Computer Engineering & Information Technology Department