Control of Robotic Manipulators

Set Point Control Technique 1: Joint PD Control

Joint torque Joint position error Joint velocity error

Why 0?

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

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:

Mixed PD Control

• Spring on endpoint and damper on joints

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

Manipulator ControlStability

Stability

• 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

Examples of responses for linear systems

Types of Stability for Nonlinear 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

Lyapunov Method

One of these is Alexandar Lyapunov, a late 1800’s Russian mathematician The other is a hipsterCan you figure out which is which?

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

Direct Approach Example: Pendulum – Find Equilibrium points

• Two equilibrium points• Stable down

• Unstable up

• Formal stability definition

Direct Approach Example: Pendulum – Find Equilibrium points

• Equations of motion

• Equilibrium points are where the time differential is 0

Math AsidePhase Planes

Phase Plane Example

• Spring-mass system with k=m=1

• Equation of motion:

• Equation of motion time response:

• Eliminating time yields:

Stability

Limit Cycles

• Linear systems = unbounded behavior

• Non-linear systems, not always the case

• EOM

• Equilibrium points

• Example of a stable limit cycle

• Other constants may lead to unstable limit cycle

Limit CycleStarting Points

Equilibrium Point

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?

Lyapunov’s Direct Method• Find some function V(x,t) such that:

Required Condition Required for:

Local Asymptotic Global

is positive definite Yes Yes Yes

is negative definite Yes -or-

is negative semi-definite Yes Yes

is negative semi-definite + Lasalle’sTheorem for time-invariant systems

Yes -or-

is negative semi-definite + Barbalat’sLemma for time-varying systems only

Yes -or-

is upper bounded (time varying systems only)

Yes Yes

V is radially unbounded Yes

Choose V

Asymptotically Globally Stable

Asymptotically Stable

Apply LaSalle’s Theorem

Apply Barbalatt’s

Lemma

Is V Positive Definite

Is 𝑉negative definite

Is V radially unbounded

Is 𝑉 negative semi-definite

Does 𝑉 = 0everywhere

Is V radially unbounded

Is the system autonomous

(time-invariant)

Locally Stable

Globally Stable

Yes

No

Yes

Yes

Yes

Yes

Yes

Yes

NoNo

No

No

No

No

Math Aside: Positive (and Negative) Definite Functions

• Example:

• Is V1 positive definite?

• Yes• V1 = 0 when x1 and x2=0?

• V1 > 0 for all x1 and x2 0.

• Example:

• Is V2 positive definite?

• No, V2 is positive semi-definite• V2=0 when x1 is 0 and x2 is

anything

• V2>0 for anything else

Mini Math Break: Positive Definite Matrices and radially unbounded ness

• A matrix, W, is positive definite if:

• A function is radially unbounded if

What do we need to be radially unbounded?

• Radially unbounded = closed contours

• Radially bounded = open contours

• Some contours (level sets) trail to infinity

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

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

Choose V

Asymptotically Globally Stable Asymptotically Stable

Apply LaSalle’s Theorem

Apply Barbalatt’sLemma

Is V Positive Definite

Is 𝑉 negative definite

Is V radially unbounded

Is 𝑉 negative semi-definite

Does 𝑉 = 0everywhere

Is V radially unbounded

Is the system autonomous (time-invariant)

Locally Stable

Globally Stable

Yes

No

Yes

YesYes

Yes

Yes

YesNo

No

No

No

No

No

• Equations of motion

• Pick

• Is V positive definite?

• Compute

• When 𝑉stops changing, what happens to the states?

• Means it approaches the origin, thus globally asymptotically stableChoose V

Asymptotically Globally Stable Asymptotically Stable

Apply LaSalle’s Theorem

Apply Barbalatt’sLemma

Is V Positive Definite

Is 𝑉 negative definite

Is V radially unbounded

Is 𝑉 negative semi-definite

Does 𝑉 = 0everywhere

Is V radially unbounded

Is the system autonomous (time-invariant)

Locally Stable

Globally Stable

Yes

No

Yes

YesYes

Yes

Yes

YesNo

No

No

No

No

No

Example: Lyapunov Stability for a pendulum with no friction

• Equation of motion

• Select

• Choose

• V = positive definite

• V is not radially unbounded

• Compute

• Expected – energy, represented by V, does not change b/c no damping

• Lasalle’s Theorem?• No• 𝑉 = 0 everywhere

• Result = locally stablePotential Energy Kinetic energy

Example: Lyapunov Stability for a damped pendulum

• EOM

• Same choice for V

• 𝑉 changes to:

• This is negative semi-definite, so the system is stable about equilibrium point, but not asymptotically stable

• 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:

• Which are the equilibrium points of the system

• Therefore, locally asymptotically stable

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 semi-definite function?

• What is the definition of a positive definite matrix?

• After you are done, compare your answers with your neighbor.

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 EnergyConservative force EnergyLike a “spring”

Lyapunov’s Direct Method

• Is V positive definite?

• Yes, if Kp and H are positive definite• H is always positive definite, can’t have negative mass or inertia

• Choose Kp to be positive definite

• Remember, if a vector A is positive definite and given any vector x, xAx>0

• Note qd is constant, so:

• Must prove that 𝑉 is negative definite

Lyapunov’s Direct Method

• Solve for H from original EOM

• Yields

?

Math Break: Skew Symmetry

• A matrix, M, is skew symmetric if:

• Therefore:

• Only one number will satisfy that, 0

• Therefore

Since it is a scalar, we can “transpose” it

Use the skew-symmetry property

Prove that is skew symmetric

• If skew symmetric then:

0

Back to Lyapunov

• Plug in the control law:

• Yields:

• Use LaSalle’s Theorem – prove that when the system’s energy does not change, the system approaches the equilibrium point

• Plug back into dynamics

• Therefore

• And

Globally Asymptotically Stable

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