Control of Robotic Manipulators

Transcription

1 Control of Robotic Manipulators

2 Set Point Control Technique 1: Joint PD Control Joint torque Joint position error Joint velocity error Why 0?

3 Equivalent to adding a virtual spring and damper to the joints

4 Technique 2: End Effector PD Control Instead of putting the spring and damper on the joints, put them in between the end effector and desired location using:

5 Mixed PD Control Spring on endpoint and damper on joints

6 PD control Mini-Quiz Write the control law for a 2D, 2-link, revolute-joint manipulator that is the equivalent to putting a damping element on the endpoint and a spring element on the joints What do you think might happen if we use the PD control law to enact trajectory control (i.e. have the end effector follow a path that is a function of time) instead of set point control? Compare you answers with your neighbor s

7 Manipulator Control Stability

8 Stability Examples of responses for linear systems When we add a control law to a system, the most fundamental question is: Is the system stable, i.e. does the response track to the desired point or trajectory?' For linear systems there are several tools that we can use to determine this without calculating the time response explicitly: Root locus Frequency response, bode plots, phase and gain margins, etc. Nyquist criteria

9 Types of Stability for nlinear Systems Local Lyapunov Stability If system starts near equilibrium, it stays near equilibrium Asymptotic Lyapunov Stability If system starts near the equilibrium, the system approaches equilibrium as time increases Exponential Lyapunov Stability If system starts near the equilibrium, the system approaches equilibrium at an exponential rate Global Asymptotic or Exponential Lyapunov Stability Independent of where the system starts, it approaches the equilibrium Unstable

10 Lyapunov Method One of these is Alexandar Lyapunov, a late 1800 s Russian mathematician The other is a hipster Can you figure out which is which?

11 Two Methods Linearization Linearize the system about the equilibrium point Lyapunov s method provides theoretical framework for linear control Direct or second Method Applicable to non linear systems The key idea is that we can consider a system to be stable if: when it starts near an equilibrium point, it stops near the equilibrium point. Vague word

12 Direct Approach Example: Pendulum Find Equilibrium points Two equilibrium points Stable down Unstable up Formal stability definition

13 Direct Approach Example: Pendulum Find Equilibrium points Equations of motion Equilibrium points are where the time differential is 0

14 Math Aside Phase Planes

15 Phase Plane Example Spring-mass system with k=m=1 Equation of motion: Equation of motion time response: Eliminating time yields:

16 Stability

17 Limit Cycles Linear systems = unbounded behavior n-linear systems, not always the case EOM Starting Points Limit Cycle Equilibrium points Example of a stable limit cycle Other constants may lead to unstable limit cycle Equilibrium Point

18 Mini-Quiz Write the equations of motion for a mass-spring-damper system Find the equilibrium point(s) What are the different classifications of Lyapunov stability?

19 Lyapunov s Direct Method Find some function V(x,t) such that: Required Condition Required for: Local Asymptotic Global is positive definite is negative definite -oris negative semi-definite is negative semi-definite + Lasalle s Theorem for time-invariant systems -oris negative semi-definite + Barbalat s Lemma for time-varying systems only -oris upper bounded (time varying systems only) V is radially unbounded

20 Choose V Is V Positive Definite Is V negative definite Is V radially unbounded Is V negative semi-definite Does V = 0 everywhere Globally Stable Is the system autonomous (time-invariant) Apply LaSalle s Theorem Apply Barbalatt s Lemma Locally Stable Asymptotically Globally Stable Is V radially unbounded Asymptotically Stable

21 Math Aside: Positive (and Negative) Definite Functions Example: Example: Is V 1 positive definite? V 1 = 0 when x 1 and x 2 =0? V 1 > 0 for all x 1 and x 2 0. Is V 2 positive definite?, V 2 is positive semi-definite V 2 =0 when x 1 is 0 and x 2 is anything V 2 >0 for anything else

22 Mini Math Break: Positive Definite Matrices and radially unbounded ness A matrix, W, is positive definite if: A function is radially unbounded if

23 What do we need to be radially unbounded? Radially unbounded = closed contours Radially bounded = open contours Some contours (level sets) trail to infinity

24 Intuition for Lyapunov s Theorem Consider a second order system Let V be positive definite If V always decreases, then it must eventually reach 0 For a stable system, all trajectories must move so that the values of V are decreasing Computing V couples the Lyapunov function to the system dynamics: Vmust be negative definite to have V approach 0 instead of infinity

25 LaSalle s Invariance Theorem Difficult to get V < 0(negative definite) Proves asymptotic stability Usually V 0(negative semi-definite) Means the system is stable, but not asymptotically stable Applies to autonomous (not time-invariant) or periodic systems

26 Choose V Is V Positive Definite Is V negative semi-definite Does V = 0 everywhere Is the system autonomous (time-invariant) Is V negative definite Is V radially unbounded Globally Stable Apply LaSalle s Theorem Apply Barbalatt s Lemma Locally Stable Asymptotically Globally Stable Is V radially unbounded Asymptotically Stable

27 Equations of motion Compute Pick Is V positive definite? Choose V Is V Positive Definite Is V negative definite Is V negative semi-definite Does V = 0 everywhere Is V radially unbounded When Vstops changing, what happens to the states? Means it approaches the origin, thus globally asymptotically stable Globally Stable Is the system autonomous (time-invariant) Apply LaSalle s Theorem Apply Barbalatt s Lemma Locally Stable Asymptotically Globally Stable Is V radially unbounded Asymptotically Stable

28 Example: Lyapunov Stability for a pendulum with no friction Equation of motion V is not radially unbounded Compute Select Choose Potential Energy Kinetic energy Expected energy, represented by V, does not change b/c no damping Lasalle s Theorem? V = 0 everywhere Result = locally stable V = positive definite

29 Example: Lyapunov Stability for a damped pendulum EOM Same choice for V V changes to: We know that is not true The system should be asymptotically stable LaSalle s Theorem: when the system s energy doesn t change, does the system state go to an equilibrium point? From the dynamics: This is negative semi-definite, so the system is stable about equilibrium point, but not asymptotically stable Which are the equilibrium points of the system Therefore, locally asymptotically stable

30 Lyapunov Mini Quiz What do you need to prove to ensure a system is globally asymptotically stable? What are the definitions of a negative definite and negative semidefinite function? What is the definition of a positive definite matrix? After you are done, compare your answers with your neighbor.

31 Example: Prove PD control law is stable for 2 link planar manipulator Equations of motion: Where: Choose total energy of system (dynamics + control law) Look at control law: Represents a conservative force. Include in total energy Represents a non-conservative force. Do not include in total energy Inertial Kinetic Energy Conservative force Energy Like a spring

32 Lyapunov s Direct Method Is V positive definite?, if K p and H are positive definite H is always positive definite, can t have negative mass or inertia Choose K p to be positive definite Remember, if a vector A is positive definite and given any vector x, xax>0 te q d is constant, so: Must prove that V is negative definite

33 Lyapunov s Direct Method Solve for H from original EOM Yields?

34 Math Break: Skew Symmetry A matrix, M, is skew symmetric if: Therefore: Since it is a scalar, we can transpose it Use the skew-symmetry property Only one number will satisfy that, 0 Therefore

35 Prove that is skew symmetric If skew symmetric then: 0

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37 Back to Lyapunov Plug in the control law: Plug back into dynamics Yields: Use LaSalle s Theorem prove that when the system s energy does not change, the system approaches the equilibrium point Therefore And Globally Asymptotically Stable

38 Lyapunov Mini-Quiz Write your name on a piece of paper and answer the following questions What is a skew-symmetric matrix? What is the difference between set-point control and trajectory control? Did we prove stability for a set-point or trajectory PD controller? After you are done, compare your answers with a neighbor s