1. Manipulator Dynamics
Amirkabir University of Technology Computer Engineering & Information Technology Department

2. Introduction
Robot arm dynamics deals with the mathematical formulations of the equations of robot arm motion.
They are useful as:
An insight into the structure of the robot system.
A basis for model based control systems.
A basis for computer simulations.

3. Equations of Motion
The way in which the motion of the manipulator arises from torques applied by the actuators, or from external forces applied to the manipulator.

4. Forward and Inverse Dynamics
Given a trajectory point, and
find the required vectors of joint torques,
Given a torque vector,
calculate the resulting motion of the manipulator,
and
: problem of controlling the manipulator
: problem of simulating the manipulator

5. Two Approaches Energy based: Lagrange-Euler. Simple and symmetric.
Momentum/force approach:Newton-Euler.
Efficient, derivation is simple but messy, involves vector cross product. Allow real time control.

6. Newton-Euler Algorithm
Newton-Euler method is described briefly below. The goal is to provide a big picture understanding of these methods without getting lost in the details.

7. Newton-Euler Algorithm
Newton-Euler formulations makes two passes over the links of manipulator
Velocities, Accelerations
Forces, moments
Gravity

8. Newton-Euler Algorithm
Forward computation
First compute the angular velocity, angular acceleration, linear velocity, linear acceleration of each link in terms of its preceding link.
These values can be computed in recursive manner, starting from the first moving link and ending at the end-effector link.
The initial conditions for the base link will make the initial velocity and acceleration values to zero.

9. Newton-Euler Algorithm
Backward computation
Once the velocities and accelerations of the links are found, the joint forces can be computed one link at a time starting from the end-effector link and ending at the base link.

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

11. 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

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

13. 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.

14. 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:

15. 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:

16. 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.

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

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

19. Simultaneous linear and rotational velocity
We skip 5.4!

20. 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.

21. 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

22. 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.

23. 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.

24. Velocity Vectors of Neighboring Links

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

26. 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.

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

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

29. 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.

30. 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

31. Example 5.3
Frame assignments for the two link manipulator

32. Example 5.3
We compute link transformations:

33. Example 5.3
Link to link transformation

34. Example 5.3
Velocities with respect to non moving base

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

36. 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?

37. Acceleration of a Rigid Body
Linear and angular accelerations:

38. Linear Acceleration : origins are coincident. : re-write it as.
: by differentiating.

39. Linear Acceleration the case in which the origins are not coincident
: when is constant
: the linear acceleration of the links of a manipulator with
rotational joints.

40. Angular Acceleration
B is rotation relative to A and C is rotating relative to B
: the angular acceleration of the links of a manipulator.

41. Inertia
If a force acts of a body, the body will accelerate. The ratio of the applied force to the resulting acceleration is the inertia (or mass) of the body.
If a torque acts on a body that can rotate freely about some axis, the body will undergo an angular acceleration. The ratio of the applied torque to the resulting angular acceleration is the rotational inertia of the body. It depends not only on the mass of the body, but also on how that mass is distributed with respect to the axis.

42. Mass Distribution
Inertia tensor- a generalization of the scalar moment of inertia
of an object

43. Moment of Inertia The moment of inertia of a solid body with density
w.r.t. a given axis is defined by the volume integral
where r is the perpendicular distance from the axis of
rotation.

44. Moment of Inertia This can be broken into components as:
for a discrete
distribution of mass
for a continuous
distribution of mass

45. Moment of Inertia The inertia tensor relative to frame {A}:
Mass moments of inertia
Mass products of inertia

46. Moment of Inertia
If we are free to choose the orientation of the reference frame, it is possible to cause the products of inertia to be zero.
Principal axes.
Principal moments of inertia.

47. Example 6.1
{C}

48. Parallel Axis Theorem
Relates the inertia tensor in a frame with origin at the center of mass to the inertia tensor w.r.t. another reference frame.

49. Measuring the Moment of Inertia of a Link
Most manipulators have links whose geometry and composition are somewhat complex. A pragmatic option is to measure the moment of inertia of each link using an inertia pendulum.
If a body suspended by a rod is given a small twist about the axis of suspension, it will oscillate with angular harmonic motion, the period of which is given by.
where k is the torsion constant of the suspending rod
, i.e., the constant ratio between the restoring torque
and the angular displacement.

50. Newton’s Equation
Force causing the acceleration

51. Euler’s Equation
Moment causing the rotation

52. Iterative Newton-Euler Dynamic Formulation
Outward iterations to compute velocities and accelerations
The force and torque acting on a link
Inward iterations to compute forces and torques

53. The Force Balance for a Link

54. The Torque Balance for a Link

55. Force Balance Using result of force and torque balance:
In iterative form:

56. The Iterative Newton-Euler Dynamics Algorithm
1st step:
Link velocities and accelerations are iteratively computed from link 1 out to link n and the Newton-Euler equations are applied to each link.
2nd step:
Forces and torques of iteration and joint actuator torques are computed recursively from link n back to link 1.

57. Outward iterations

58. Inward iterations

59. Inclusion of Gravity Forces
The effect of gravity loading on the links can be included by setting , where G is the gravity vector.

60. The Structure of the Manipulator Dynamic Equations
: state space equation
: mass matrix
: centrifugal and Coriolis terms
: gravity terms
: configuration space
: matrix of Coriolis coefficients
: centrifugal coefficients

61. Coriolis Force
A fictitious force exerted on a body when it moves in a rotating reference frame.

62. Lagrangian Formulation of Manipulator Dynamics
An energy-based approach (N-E: a force balance approach)
N-E and Lagrangian formulation will give the same equations of motion.

63. Kinetic and Potential Energy of a Manipulator
Total kinetic energy of a manipulator
Total potential energy of a manipulator

64. Lagrangian
Is the difference between the kinetic and potential energy of a mechanical system

65. The equations of motion for the manipulator
vector of actuator torque

66. Example 6.5
: variable
The center of mass of link 1 and link 2

67. Manipulator Dynamics in Cartesian Space
Joint space formulation
Cartesian space formulation

68. Expressions for the terms in the Cartesian dynamics:

69. The Cartesian configuration space torque equation:
:Coriolis coefficients
:Centrifugal coefficients

70. Dynamic Simulation: (Euler Integration)
Simulation requires solving the dynamic equation for acceleration
Nonrigid body effects: friction
: Given initial conditions
We apply numerical integration to compute positions and velocities:

71. Trajectory Generation
Next Course:
Trajectory Generation
Amirkabir University of Technology Computer Engineering & Information Technology Department