1. Chapter 7 Inverse Dynamics Control

2. Table of Contents Forward dynamics and Inverse dynamics PD Control
Computed Torque Method
Inverse Dynamic control
Realization of Robust Control

3. 7.1 Forward dynamics and Inverse dynamics
Given input torque , the joint trajectories will be obtained.
The joint-trajectories are calculated by the nonlinear differential equations.
Dynamics problem:
Suppose is twice differentiable function, it will be obtained input torque
for joint trajectory

4. Dynamics equations In a given trajectory , dynamics equations are
: Inertial terms according to the mass distribution
: Centrifugal-force/ Coriolis-force
: Gravitation
: Torque
(7.1)

5. Relationship of Forward & Inverse Dynamics
[Fig. 7-1] Relationship of Forward & Inverse Dynamics

6. Relationship of Forward & Inverse Dynamics
[Fig. 7-2] Relationship of Forward & Inverse Dynamics

7. 7.2 PD Control Dynamics Model of Robot (including motor dynamics)
(7.2)
PD Control Input
(7.3)
assuming g(q) is given…

8. PD Control System Block diagram
[Fig. 7-3] PD Control System
: Subtraction of the desired joint-displacement and actual joint-displacement
: Proportional & differential gains

9. Lyapunov function V
In order to determine the stability of the system, a Lyapunov function is defined as
First term : Kinetic energy of the robot
Second term : Proportional Feedback,
is zero where “target point” , otherwise is a positive function.
(7.4)

10. Time differential of V
Along any motion of robot, the function is decreasing to zero.
- This will imply that the robot is moving toward the desired goal
configuration.
Time derivative of
(7.5)

11. Time differential of V
Solving for in Eq. (7.2) with known and substituting the resulting expression into Eq. (7.5) yields
(7.6)
where : skew symmetric,

12. PD Control law is decreasing as long as is not zero.
(7.7)
is decreasing as long as is not zero.
Suppose Then Eq. (7.7) implies that and
hence
(7.8)
LaSalle’s theorem : The system is asymptotically stable.

13. Definition diagonal elements of the matrix are all zero
pf )
(7.9)
is a condition for the
Thus, the diagonal matrix of is zero.

14. Theorem of LaSalle Consider a nonlinear system on .
Suppose a Lyapunov function candidate is found such that, along
solution trajectories
Asymptotically stable if does not vanish ideally along any solution
of Eq. (7.10) other than the null solution.
Eq. (7.10) is asymptotically stable if the only solution of Eq. (7.10)
satisfying
(7.10)

15. Theorem of LaSalle
The unknown gravitational terms presented in Eqs. (7.2) and (7.6) can be modified to
The presence of the gravitational term in Eq. (7.11) means that PD control alone can not guarantee asymptotic tracking.
In practice, there will be a steady state error or offset.
Assuming that the closed loop system is stable at the robot configuration .
That can be achieved by satisfying
(7.11)
(7.12)
(7.13)

16. Theorem of LaSalle
The steady state “holding torque” is sufficient to balance the gravitational torque
The steady state error can be reduced by increasing the position gain
In order to remove this steady state error, the PD control law can be modified as
(7.14)

17. 7.3 Computed Torque Method
Computed Torque Method is a control method to simplify a nonlinear
system to a linear system.
Computed torque method is used for robust, adaptive and learning controls.
[Fig. 7-4] Block diagram of computed torque method

18. Linear System The control input U is applied to the system:
From Eqs. (7.15) and (7.16), the system becomes linear as
(7.15)
(7.16)
(7.17)

19. Linear System
In general, as shown in Figure 7-4, the PD control is added to the outer loop
feedback. The input of block, is represented as following:
where
(7.18)
Substitution Eq. (7.18) into Eq. (7.17)
(7.19)

20. 7.4 Inverse Dynamic Control
n-link robot equation of motion from Eq. (7.2)
where,
(7.20)

21. Nonlinear feedback control law
The idea of inverse dynamics is to seek a nonlinear feedback control law:
Substituting this into Eq. (7.20), it results in a linear closed loop system.
(7.21)

22. Inverse Dynamic control
With the equation of motion of manipulator from Eq. (7.20),
choose the control input as
Since the inertia matrix is invertible, the combined system
Eq. (7.20) is reduced to
The term represents a new input to the system which needs to be
chosen. Eq. (7.23) is known as the double integrator system as it
represents n uncoupled double integrators.
(7.22)
(7.23)

23. Inverse Dynamic control
Assuming that is a function only of and its derivatives, then
will affect the independently of the motion of the other links.
Since can now be designed to control a simple linear 2nd order system, the obvious choice is to set.
  and are diagonal matrices with diagonal elements consisting
of position and velocity gains, respectively. The closed loop system
is then the linear system as
(7.24)
(7.25)

24. Inverse Dynamic Control
When the trajectory follows the equation
Reference input is represented as
The error is satisfied under the equation as
(7.26)
(7.27)
(7.28)

25. Inverse Dynamic control
Gain matrices ,
result in a closed loop system which is globally decoupled, with each joint response equal to the response of a critically damped linear second order system with natural frequency .
The natural frequency determines the speed of response of the joint.

26. Acceleration, Dynamics of Manipulator
is invertible for We may solve for the acceleration of the manipulator as
The dynamics of the manipulator, which is a position control device, would be given as
: the input acceleration vector.
This is again the familiar double integrator system.
(7.29)
(7.30)

27. Acceleration, Dynamics of Manipulator
Eq. (7.30) is not an approximation in any sense, rather it represents the actual
open loop dynamics of the system, where the acceleration is chosen as the
input.
Our physical universe prohibits such “acceleration actuators” and we must
be content with the ability to produce a generalized force(torque) at each
joint .
Comparing Eq. (7.29) and (7.30), we see that the torque and the
acceleration of the manipulator are related by
(7.31)

28. Input conversion
By the invertibility of the inertia matrix, we may solve for the input
torque as
which is the same as the previously derived expression (7.22).
Thus the inverse dynamics can be viewed as an input transformation which transforms the problem from one of choosing acceleration input commands.
(7.32)

29. Inner Loop/Outside Loop Control Structure
Implementation of Control System
[Fig. 7-5] Inner loop/outside loop control structure

30. Outside Loop Control
Figure 7-5 means that the computation of the nonlinear control is performed
in an inner loop, perhaps with a dedicated hardware interface, with the vectors
and as its inputs and as output.
The outer loop in the system is then the computation of the additional input
term .
Note that the outer loop control is more close to a feedback control in the
usual sense.

31. Problem
The method of inverse dynamics is thus very attractive from a control standpoint since the highly nonlinear coupled dynamics of the manipulator are canceled and replaced by a simple decoupled linear second order system.
However, such exact cancellation schemes leave many issues of sensitivity and robustness that must be addressed.

32. 7.5 Implementation and robustness issues
The input of control, Eq. (7.22) is approximately as follows:
The uncertainty and modeling error:
Extremely, it can be changed as
(7.33)
(7.34)
(7.35)

33. Robust control If the input is equal to the torque value, it becomes
and
If we assume
(7.36)
(7.37)

34. Uncertainty The above equation is represented w.r.t the error, as
where , hereafter called the uncertainty, is given by the expression.
In here, E is defined as
(7.38)
(7.39)

35. Example 1
Consider the single-link robot of the figure below, modeled for simplicity
by the equation
(7.40)
[Fig. 7-6] Single-link robot example

36. Inverse dynamic control
where I is the total moment of inertia about the joint axis, M is the total mass, and L is the distance from the joint axis to the center of mass of the system.
The inverse dynamics control law is
Suppose that the coefficients I and MgL are unknown but that
(7.41)

37. Cont. If we choose the control law
with and then the response for a given desired trajectory is shown in the following Fig.7-7.
(7.42)

38. Tracking response Output :
[Fig. 7-7] Approximate inverse dynamics control

39. Unstability can't ensure a stability as it is a function of .
So, pursuing a stability by addition of like the following:
(7.43)

40. Worst case bound Assumptions of the worst case for the robot
(7.44)
(7.45)
(7.46)

41. Assumption 2 Assume bound of value is determined,
(7.47)
Since the inertia matrix is uniformly positive definite for all , there exist positive constants such that
is defined as following
(7.48)

42. Stabilizing control v
Notice that there is always at least one choice of satisfying
assumption 2.
becomes a small value by determining which is considered
as inertia parameter and load.
(7.49)

43. Example. If it is defined as and ,
it can be displayed as following state:
In here,
(7.51)

44. Outer loop term v (Step 1)
Step 1: Since the matrix A in Eq. (7.51) is unstable,
we set v as
Define as , given as following.
In here, it is set as ,
(7.52)
(7.53)

45. Outer loop term v
Extend ,
, instead of
(7.54)
(7.55)

46. Outer loop term v system like the following:
If it is expressed as vector form, it equals time varying non-linear
system like the following:
This form is to be Hurwitz as (all eigenvalues have negative real parts). In Eq. (7.51),
If is assigned instead of v
where
(7.56)
(7.57)

47. Uncertainty compensation for (Step 2 )
The ideal case, , the Inverse Dynamic Control can be used.
Step 2: A continuous function is defined for Hurwitz
system to satisfy the following inequalities:
(7.58a)
(7.58b)

48. Uncertainty compensation for
The decision of :
where
(7.59)
If we solve for , it equals like

49. Step 3 Step 3: Since is Hurwitz, choose a symmetric,
positive definite matrix Q , then choose the unique positive definite matrix P satisfied to the Lyapunov equation.
(7.60)

50. Step 4 Step 4: to outside loop controller be setting as follows:
satisfying Eq.(7.58a)
Stable control system can be obtained
(7.61)
(7.62)

51. Prove
, needs to be proved.
In Eq. (7.58b)
(7.63)

52. Prove
For simplicity set
If ,
If , it is set as
(7.64)

53. Prove So, Eq. (7.64) can be written as because,
As the whole, Eq. (7.63) becomes negative.
Since the control is non-linear, the chattering occurs.
(7.65)

54. Proof of Lyapunov Equation
(7.66)
is positive definite It is proved that of positive definite
satisfying Eq. (7.66) exists.
1. It shows possibility of existence of P satisfying Eq. (7.66) for a
positive definite .
The definition of matrix,
Prove

55. Positive definite ⇒ proof of existence
2. If is positive definite, P which satisfying Eq.(7.66) is positive definite.
(7.67)
⇒ proof of existence

56. Positive definite , when , when is positive definite, , the system is
(7.68)
, when is positive definite,
, the system is
P is positive definite.

57. Uniqueness 3. Uniqueness For a , it is assumed that two exist: So,
(7.69)
, constant for all t
(7.70)

58. Uniqueness ⇒ Left is the value, when Right is the value, when
it should be
i.e., ,
is unique.

59. Example 2
We will illustrate the above approach to robust design by providing a detailed treatment of the system of Example 1. With the equation of motion
We choose the control law
where
(7.71)
(7.72)

60. Cont. is to be designed according to Eq.(7.61).
In terms of the tracking error
We can write Eq.(7.71) and Eq.(7.72) together as
(7.73)

61. Cont.
If ,
We have
Finally,
(7.74)
(7.75)

62. Cont.
With worst case bounds
choosing
Estimation value
(7.76)

63. Cont.
Therefore,
Set Q=I and solve the Lyapunov equation
(7.77)

64. Cont.
If ,

65. Cont. The unique positive definite solution is Set Choose according to
(7.78)

66. Tracking response with additional compensation
Output :
[Fig. 7-8] Tracking response with additional compensation

67. Tracking errors with and without additional compensation
[Fig. 7-9] Tracking errors with and without additional compensation

68. Homework
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