control of robotic manipulators - the robotics lab@iit -...
TRANSCRIPT
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Control of Robotic Manipulators
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Set Point Control Technique 1: Joint PD Control
Joint torque Joint position error Joint velocity error
Why 0?
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• Equivalent to adding a virtual spring and damper to the joints
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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:
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Mixed PD Control
• Spring on endpoint and damper on joints
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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
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Manipulator ControlStability
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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
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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
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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?
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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
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Direct Approach Example: Pendulum – Find Equilibrium points
• Two equilibrium points• Stable down
• Unstable up
• Formal stability definition
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Direct Approach Example: Pendulum – Find Equilibrium points
• Equations of motion
• Equilibrium points are where the time differential is 0
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Math AsidePhase Planes
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Phase Plane Example
• Spring-mass system with k=m=1
• Equation of motion:
• Equation of motion time response:
• Eliminating time yields:
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Stability
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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
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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?
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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
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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
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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
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Mini Math Break: Positive Definite Matrices and radially unbounded ness
• A matrix, W, is positive definite if:
• A function is radially unbounded if
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What do we need to be radially unbounded?
• Radially unbounded = closed contours
• Radially bounded = open contours
• Some contours (level sets) trail to infinity
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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
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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
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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
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• 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
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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
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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
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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.
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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”
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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
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Lyapunov’s Direct Method
• Solve for H from original EOM
• Yields
?
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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
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Prove that is skew symmetric
• If skew symmetric then:
0
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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
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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