control of robot manipulators professor nicola ferrier me room 2246, 265-8793 [email protected]
TRANSCRIPT
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Control
controller
Robot Level task
TrajectoryPlanning(IK, J, etc)
Pe(t)
Tasks
Powerelectronics Current to
motors
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Control
controller
Robot Level task
TrajectoryPlanning(IK, J, etc)
Pe(t)
Tasks
Powerelectronics Current to
motors
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Control
• Independent Joint Control (chapter 6*)– Use computed reference points (setpoints) for
each joint– Control each joint “independently”
• Ignore dynamic effects• Treat each joint as a stand alone “motor”
• Dynamics Based control (Chapter 8*)– Use dynamics model to facilitate control
• Compute torque feedforward• Inverse Dynamic Control • Operation Space control• And Compliance, Impedance, Force….
*Spong, Hutchinson, Vidyasagar, “Robot Modeling and Control”, Wiley, 2006
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Jointed system components
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Independent Joint Control
• Use computed reference points (setpoints) for each joint
• Control each joint “independently”– Ignore dynamic effects– Treat each joint as a stand alone “motor”
• Simplifies control• Block Diagram (next slide)
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Block Diagram of PE controller for a single joint
Joint position sensor
PE controlleru = Kpe e
Motor +reduction
+transmissionJoint
Reference angle
Error signal
control signal torque
Actual joint position, q
measured joint
position, qm
Energy source
(current)
u
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Independent Joint Control
• Control each joint independently without “communication” between actuators
• Basic Steps:– Model actuator– Use kinematics to obtain set-points for each
joint– Develop a controller for each joint– Error for joint i:
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Actuator Model
• Need to model relationships:– between actuator input (current) and output
(torque)• Section 6.1: Permanent magnet DC-motor
– Torque is approximately linear with applied current» (equation 6.4)
Motor torque, NmApplied current, amp
Motor constant, Nm/amp
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actuator current vs torque
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Actuator Model
• Need to model relationships:– between actuator torque and motor angle (q)
• Section 6.1: Permanent magnet DC-motor – Second order ode
» (equation 6.8 and 6.16)
Rotational inertia of joint, kg m^2
control input
Effective damping (friction, back emf), Nm/amp
disturbance
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Independent Joint Control
• Control each joint independently without “communication” between actuators
• Basic Steps:Model actuatorUse kinematics to obtain setpoints for each
joint (recall trajectory planning – chapter 5)– Develop a controller for each joint
• Error for joint i:
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Proportional control for each joint
• Input proportional to position error:
• Neglect disturbance, wlog set reference position to zero
• or
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Proportional control for each joint
• Second order linear differential equation:
• has general form solution:
– where
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Block Diagram of PE controller
+-
Kpe
Sensor transfer function
s(Js+F)1
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Three solutions
What does this term do?
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Three solutions
• Over-damped (2 > 0)
• Critically damped (2 = 0)
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Three solutions
• Under-damped (2 < 0)– has complex roots
– Oscillates with frequencyIf B is small and
KPE is large: unstable!
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Example Step Responses (1 radian)
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PI, PID controllers
• PE controllers can lead to – Steady state error– Unstable behavior
• Add Integral Term:
….but now we can have overshoot• Add derivative term (PID Controller)
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Block Diagram of PE controller
+-
Kpe
Sensor transfer function
s(Js+F)1
Ki(1/s)
Kd(s)
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Set Gains for PID Controller
• wlog set (we already have )
• Convert to third order equation
• Solution will be of the form
– where
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Set Gains for PID Controller
• Critically damped when = 0 or
• An equation in 3 unknowns• Need two more constraints:
– Minimum energy– Minimum error– Minimum jerk
• And we need the solution to double minimization– Beyond the scope of this class – topic of optimal
control class
More on gains in Advanced Robotics Course ….
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Problems with Independent Joint Control
• Synchronization ?– If one joint does not follow the trajectory,
where is the end-effector???
• Ignores dynamic effects – Links are connected– Motion of links affects other links– Could be in-efficient use of energy
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Dynamics: Equations of Motion
• Dynamic Model– forces/torques motion of manipulator+load
• Equations of Motion
• Ideally we can use
– Modeling errors– Friction– synchronization