1. Forward Kinematics

2. DH Representation

3. DH Representation - Frames
(Convention) Frame i is attached to link i. The inertial frame is Frame 0 and Earth is link 0. Joint i joins links i-1 and i.
(Another convention) The joint i+1 rotates about axis zi
(DH1) The axis xi is perpendicular to the axis zi-1
(DH2) The axis xi intersects the axis zi-1
DH convention imposes two constrains thus enabling the use of only four parameters instead of six.

4. DH1

5. DH2

6. Assign frames at the two joints and at the end-point based on DH convention
Write A1, A2, and A3
Frame i is attached to link i.
The inertial frame is Frame 0 and Earth is link 0.
Joint i joins links i-1 and i.
The joint i+1 rotates about axis zi
Link ai αi di 1 2

8. Three link cylindrical robotai αi di i 1 2 3

10. Stanford Manipulator
Link ai αi di i 1 2 3 4 5 6

11. SCARA Manipulator
Link ai αi di i 1 2 3 4

12. Inverse Kinematics

14. Kinematic Decoupling
Last three joints intersecting at a point (spherical wrist)
Inverse position kinematics
Inverse orientation kinematics
Let

15. Find q1, q2, q3 such that is located at o.
With q1, q2, q3 obtain
Solve for q1, q2, q3 using the expression for from forward kinematic equations and its numerical value obtained from

17. Euler Angles

18. Velocity Kinematics Angular velocity of frame 2 wrt frame 0
velocity of p wrt frame 0

21. Angular Velocity

23. Linear Velocity

24. Velocity Kinematics - 2 DOF

25. Velocity Kinematics - 2 DOF

26. Singularities
The values of q for which the rank of J decreases are called the singularities or singular configurations.

27. Singularities 2-DOF Velocity can be given in only x and y direction.
when theta2 is zero. In that configuration arbitrary velocity in x and y direction cannot be imparted. One example is velocity in line with the manipulator when theta2 is zero.

28. Importance of Singularities
certain direction of motion may be unattainable.
bounded gripper velocities to unbounded joint velocities.
bounded gripper forces and torques may correspond to unbounded joint torques.
usually points on the boundary of the workspace.
unreachable points in workspace under small perturbation of link parameters such as length, offset, etc.
non-unique solution to inverse kinematics problem.

29. Inverse kinematics and Jacobian
For desired end-point position and orientation as a function of time (the usual robot operation) X-dot can be evaluated and by inverting the Jacobian and so by integrating from a known initial value of q, other values of q can be obtained for given positions and orientations.
This method is great but for singularities of the Jacobian.

30. Dynamics
Differential equation relating input torques and forces to the positions (angles) and their derivatives.
Like force = mass times acceleration.

31. Euler-Lagrange Equations
Equations of motion for unconstrained system of particles is straightforward (F = m x a).
For a constrained system, in addition to external forces, there exist constrained forces which need to be considered for writing dynamic equations of motion.
To obtain dynamic equations of motion using Euler-Lagrange procedure we don’t have to find the constrained forces explicitly.

32. Holonomic Constraints

33. Nonholonomic Constraints
General expression for nonholonomic constraints is:
Nonholonomic constraints contain velocity terms which cannot be integrated out.
A rear powered front steering vehicle

34. Consider a system of k particles, with corresponding coordinates,
Often due to constraints or otherwise the position of k particles can be written in terms of n generalised coordinates (n < k),
In this course we consider only holonomic constraints and for those constraints one can always find in principle n (n < k) independent generalised coordinates.

35. Virtual Displacements
Define virtual displacements from above by setting dt=0.
In our case dqj are independent and satisfy all the constraints. If they additional constraints have to be added to dqj to finally arrive at a statement of virtual displacements which have only independent dqj, we can replace dqj with

36. Virtual Work
Let Fi be total force on every particle, then virtual work is defined as:
Constraining forces do no work when a virtual displacement takes place (as is the case with holonomic constraints), so in equilibrium

37. D’Alembert’s Principle
D’Alembert’s principle states that, if one introduces a fictitious additional force, the negative of the rate of change of particle i, then each particle will be in equilibrium.

38. Generalised Forces
is called the i-th generalised force. The equations of motion become:

41. Euler-Lagrange Equations of Motion

44. Write the dynamic equations of motions for this system.

46. Expression for Kinetic Energy

47. Attach a coordinate frame to the body at its centre of mass, then velocity of a point r is given by:

50. Frame for I and Omega
The expression for the kinetic energy is the same whether we write it in body reference frame or the inertial frame but it is much easier to write I in body reference frame since it doesn’t change as the body rotates but its value in the inertial frame is always changing.
So we write the angular velocity and the inertia matrix in the body reference frame.

51. Jacobian and velocity
D(q) is a symmetric positive definite matrix and is known as the inertia matrix.

52. Potential Energy V

53. Two Link Manipulator
Links are symmetric, centre of mass at half the length.