COM2009-3009 Robotics: Lecture 6 PDF

© 2023 The University of Sheffield COM2009-3009 Robotics COM2009-3009 Robotics: Lecture 6 slide 1 1 © 2023 The University of Sheffield COM2009-3009 Robotics Lecture 6 PID Control COM2009-3009 Robotics: Lecture 6 slide 2 2 1 © 2023 The University of Sheffield Negative-Feedback Control D...

© 2023 The University of Sheffield COM2009-3009 Robotics COM2009-3009 Robotics: Lecture 6 slide 1 1 © 2023 The University of Sheffield COM2009-3009 Robotics Lecture 6 PID Control COM2009-3009 Robotics: Lecture 6 slide 2 2 1 © 2023 The University of Sheffield Negative-Feedback Control DiStefano III, J. J., Stubberud, A. R., & Williams, I. J. (1990). Feedback and Control Systems (2nd ed.). New York: McGraw-Hill. COM2009-3009 Robotics: Lecture 6 slide 3 3 © 2023 The University of Sheffield Negative-Feedback Control Proportional Integral Derivative (PID) Controller 𝑢 𝑡 = 𝐾! 𝑒 𝑡 + 𝐾" ' 𝑒 𝑡 𝑑𝑡 + 𝐾# COM2009-3009 Robotics: Lecture 6 𝑑𝑒 𝑡 𝑑𝑡 slide 4 4 2 © 2023 The University of Sheffield PID Control Proportional 𝐾!𝑒 𝑡 + 𝑟 𝑡 ∑ + 𝑒 𝑡 Integral 𝐾" ∫ 𝑒 𝑡 𝑑𝑡 - + ∑ 𝑢 𝑡 Actuator + Derivative $% 𝐾# $& 𝑡 Sensor 𝑢 𝑡 = 𝐾! 𝑒 𝑡 + 𝐾" ' 𝑒 𝑡 𝑑𝑡 + 𝐾# COM2009-3009 Robotics: Lecture 6 𝑑𝑒 𝑡 𝑑𝑡 slide 5 5 © 2023 The University of Sheffield History of PID Control Proportional • 1911: first developed Elmer Sperry • 1933: the Taylor Instrumental Company (TIC) introduced the first pneumatic controller with a fully tuneable proportional controller • 1930s: control engineers discovered that they could eliminate the steady state error found in proportional controllers by integrating, giving rise to the proportionalintegral controller • 1940: TIC developed the first pneumatic controller with a derivative action to reduce overshooting issues • 1942: Ziegler-Nichols tuning rules introduced • 1950s: PID controllers became widely adopted in industry "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% - + ∑ + Derivative ") *+ *, % . % Actuator Sensor COM2009-3009 Robotics: Lecture 6 slide 6 6 3 © 2023 The University of Sheffield PID Control • The three ‘gain’ terms have different effects … Proportional "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% + - ∑ . % Actuator + *+ ") *, % Derivative Sensor – KP influences the short-term error – KI influences the long-term error – KD influences the rate-of-change of error • Control is possible using … – KP only (known as a ‘P’ controller) – KP and KI (known as a ‘PI’ controller) – KP, KI and KD (a full ‘PID’ controller) • Optimal control requires appropriate values for KP, KI and KD 𝑢 𝑡 = 𝐾! 𝑒 𝑡 + 𝐾" ' 𝑒 𝑡 𝑑𝑡 + 𝐾# COM2009-3009 Robotics: Lecture 6 𝑑𝑒 𝑡 𝑑𝑡 slide 7 7 © 2023 The University of Sheffield Operation of a ‘P’ Controller Proportional "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% - + ∑ . % Actuator + Derivative ") *+ *, % Sensor Reference Input error Controlled Output Time COM2009-3009 Robotics: Lecture 6 slide 8 8 4 © 2023 The University of Sheffield Operation of a ‘P’ Controller Proportional "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% + - ∑ . % Actuator + *+ ") *, % Derivative Sensor Reference Input Controlled Output Time Kp = 0 COM2009-3009 Robotics: Lecture 6 slide 9 9 © 2023 The University of Sheffield Operation of a ‘P’ Controller Proportional "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% - + ∑ . % Actuator + Derivative ") *+ *, % Sensor Reference Input Controlled Output Time Kp = too small COM2009-3009 Robotics: Lecture 6 slide 10 10 5 © 2023 The University of Sheffield Operation of a ‘P’ Controller Proportional "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% + - ∑ . % Actuator + *+ ") *, % Derivative Sensor Reference Input Controlled Output Time Kp = too large COM2009-3009 Robotics: Lecture 6 slide 11 11 © 2023 The University of Sheffield Operation of a ‘P’ Controller Proportional "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% - + ∑ . % Actuator + Derivative ") *+ *, % Sensor Reference Input Controlled Output Time Kp = wrong sign! COM2009-3009 Robotics: Lecture 6 slide 12 12 6 © 2023 The University of Sheffield Operation of a ‘P’ Controller Proportional "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% + - ∑ . % Actuator + *+ ") *, % Derivative Sensor Reference Input Controlled Output Time Kp = OK COM2009-3009 Robotics: Lecture 6 slide 13 13 © 2023 The University of Sheffield Operation of a ‘P’ Controller Proportional "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% - + ∑ . % Actuator + Derivative ") *+ *, % Sensor Reference Input Controlled Output Time Kp = better COM2009-3009 Robotics: Lecture 6 slide 14 14 7 © 2023 The University of Sheffield Quality of Control overshoot Proportional settling time "# $ % equilibrium error + - % + ∑ $ % Integral "& ∫ $ % (% + - ∑ . % Actuator + *+ ") *, % Derivative Sensor rise time Controlled Output Time COM2009-3009 Robotics: Lecture 6 slide 15 15 © 2023 The University of Sheffield Operation of a ‘PID’ Controller Proportional "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% - + ∑ . % Actuator + Derivative ") *+ *, % Sensor http://www.inpharmix.com/jps/PID_Controller _For_Lego_Mindstorms_Robots.html 𝑢 𝑡 = 𝐾! 𝑒 𝑡 + 𝐾" ' 𝑒 𝑡 𝑑𝑡 + 𝐾# COM2009-3009 Robotics: Lecture 6 𝑑𝑒 𝑡 𝑑𝑡 slide 16 16 8 © 2023 The University of Sheffield Operation of a ‘PID’ Controller Proportional "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% - + ∑ . % Actuator + Derivative *+ ") *, % Sensor https://en.wikipedia.org/wiki/PID_controller 𝑢 𝑡 = 𝐾! 𝑒 𝑡 + 𝐾" ' 𝑒 𝑡 𝑑𝑡 + 𝐾# COM2009-3009 Robotics: Lecture 6 𝑑𝑒 𝑡 𝑑𝑡 slide 17 17 © 2023 The University of Sheffield Demo: A ‘PID’ Controller 𝑢 𝑡 = 𝐾! 𝑒 𝑡 + 𝐾" ' 𝑒 𝑡 𝑑𝑡 + 𝐾# COM2009-3009 Robotics: Lecture 6 𝑑𝑒 𝑡 𝑑𝑡 slide 18 18 9 © 2023 The University of Sheffield Tuning a PID Controller 1. Manual adjustment – no maths – online – requires experience Proportional "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% - + ∑ . % Actuator + Derivative *+ ") *, % Sensor 2. Computer simulation – automatic – offline 3. The ‘Ziegler-Nichols’ Method – some maths – online COM2009-3009 Robotics: Lecture 6 slide 19 19 © 2023 The University of Sheffield The ‘Ziegler-Nichols’ Method Proportional • KP is increased from 0 until the control system has stable and consistent oscillations • This value of KP is referred to as the ‘ultimate gain’: KU • The oscillation frequency is designated as FU Hz (where TU = 1/FU secs) • Values for KP, KI and KD are calculated as follows … "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% - + ∑ . % Actuator + Derivative ") *+ *, % Sensor Ziegler, J.G & Nichols, N. B. (1942). "Optimum settings for automatic controllers". Transactions of the ASME. 64: 759–768. COM2009-3009 Robotics: Lecture 6 slide 20 20 10 © 2023 The University of Sheffield The ‘Ziegler-Nichols’ Method • KP is increased from 0 until the control system has K = 4.0 U stable and consistent oscillations • This value of KP is referred to as the ‘ultimate gain’: KU • The oscillation frequency is designated as FU (where TU = 1/FU) Ziegler-Nichols • PID-ZN FU = 1.2 Hz Values for KP, KI and KD are calculated … COM2009-3009 Robotics: Lecture 6 slide 21 21 © 2023 The University of Sheffield Alternative Formulations Proportional • Parallel (ideal) form … "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% - + ∑ . % Actuator + Derivative ") *+ *, % Sensor 𝑢 𝑡 = 𝐾! 𝑒 𝑡 + 𝐾" ' 𝑒 𝑡 𝑑𝑡 + 𝐾# 𝑑𝑒 𝑡 𝑑𝑡 • Standard (industrial) form … 1 𝑑𝑒 𝑢 𝑡 = 𝐾! 𝑒 𝑡 + ' 𝑒 𝑡 𝑑𝑡 + 𝑇# 𝑡 𝑇" 𝑑𝑡 Mathematically equivalent, but more intuitive where TI is the integral time TD is the derivative time COM2009-3009 Robotics: Lecture 6 slide 22 22 11 © 2023 The University of Sheffield Alternative Formulations 𝑢 𝑡 = 𝐾! 𝑒 𝑡 + 𝐾" ' 𝑒 𝑡 𝑑𝑡 + 𝐾# Proportional "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% - + ∑ . % 𝑑𝑒 𝑡 𝑑𝑡 Actuator + Derivative *+ ") *, % Sensor 𝑢 𝑡 = 𝐾! 𝑒 𝑡 + COM2009-3009 Robotics: Lecture 6 1 𝑑𝑒 ' 𝑒 𝑡 𝑑𝑡 + 𝑇# 𝑡 𝑇" 𝑑𝑡 slide 23 23 © 2023 The University of Sheffield Demo: PID Control in Action Maze-Runner COM2009-3009 Robotics: Lecture 6 slide 24 24 12 © 2023 The University of Sheffield PID Control in Action Proportional "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% - + ∑ . % Actuator + Derivative *+ ") *, % Sensor http://youtu.be/nYsA6tuEK3U COM2009-3009 Robotics: Lecture 6 slide 25 25 © 2023 The University of Sheffield PID Pseudo Code Maze-Runner The aim is to stay equidistant from the sides Kp = 3.6 Ki = 15.12 Kd = 0.21 speed = 50 integral = 0 last-error = 0 derivative = 0 dt = 0.01 PID While True: start = time() left-distance = READ-ULTRASOUND-SENSOR(1) right-distance = READ-ULTRASOUND-SENSOR(2) error = left-distance – right-distance integral = integral + error derivative = error – last-error turn = Kp*error + Ki*integral*dt + Kd*derivative/dt WRITE-MOTOR(1, direction=forward, power=speed+turn) WRITE-MOTOR(2, direction=forward, power=speed-turn) last-error = error end = time() dt = start - end COM2009-3009 Robotics: Lecture 6 slide 26 26 13 © 2023 The University of Sheffield The ‘Inverted Pendulum Problem’ Balance Balance-2 https://youtu.be/a4c7AwHFkT8 COM2009-3009 Robotics: Lecture 6 slide 27 27 © 2023 The University of Sheffield The ‘Inverted Pendulum Problem’ COM2009-3009 Robotics: Lecture 6 slide 28 28 14 © 2023 The University of Sheffield The ‘Inverted Pendulum Problem’ • A single PID controller is hard to calibrate • An alternative is a hierarchical arrangement of P controllers (as in Perceptual ControlTheory) – the bob’s angular position error is corrected by varying the angular velocity reference signal – the angular velocity error is corrected by varying the angular acceleration reference signal – the angular acceleration error is corrected by varying the cart’s position reference signal (with gravity providing the acceleration) – the cart’s position error is corrected by varying the cart’s linear velocity reference signal – the cart’s velocity error is corrected by varying the cart’s linear acceleration reference signal – the cart’s linear acceleration is varied by varying the motor torque COM2009-3009 Robotics: Lecture 6 slide 29 29 © 2023 The University of Sheffield The ‘Inverted Pendulum Problem’ Ben Hawker Hierarchical Perceptual Control Theory (HPCT) solution COM2009-3009 Robotics: Lecture 6 slide 30 30 15 © 2023 The University of Sheffield Compliance https://www.densowave.com/en/robot/ product/function/C ompliance-controlfunction.html • In many robot applications (e.g. handling soft/delicate materials), the actuators needs to have some ‘give’, i.e. they shouldn’t be completely stiff • However, as we have seen, a good PID controller would render an actuator highly resistant to any disturbance, hence it will be stiff by design (high Kp → high stiffness) • A low value of Kp will simply mean weak control (i.e. it might drop whatever it’s holding) • So, we want an actuator to be ‘compliant’ rather than ‘floppy’ • Also, robot actuators tend to be rigid/stiff, not just because of the controller, but because of gear wheels • Soft materials (e.g. on a gripper) have a natural compliance, but compliance can also be programmed into the controller • This can be done by sensing the actuator position and using that to modify the controller, e.g. by altering the reference/setpoint COM2009-3009 Robotics: Lecture 6 slide 31 31 © 2023 The University of Sheffield Compliance PID-2 PID-1 Semini, C, Barasuol, V. & Boaventura, T. (2016) “Is active impedence the key to a breakthrough for legged robots?”, Robotics Research, pp. 3-19. COM2009-3009 Robotics: Lecture 6 slide 32 32 16 © 2023 The University of Sheffield Compliance https://youtu.be/sbhiNNIxMNQ COM2009-3009 Robotics: Lecture 6 slide 33 33 © 2023 The University of Sheffield Limitations of PID Control • Constant parameters Proportional "# $ % + - % + ∑ $ % Integral "& ∫ $ % (% - + ∑ + Derivative ") *+ *, % Sensor . % Actuator • No direct knowledge of the process • Linear • Symmetric • Derivative term is sensitive to sensor noise • Integral term can lead to ‘integral windup’ • Single-layer solutions can be hard to optimise COM2009-3009 Robotics: Lecture 6 slide 34 34 17 © 2023 The University of Sheffield This lecture has covered … • • • • • • • • • • Negative-feedback control PID control Operation of P, PI and PID controllers Tuning a PID controller Ziegler-Nichols Alternative formulations PID pseudo code The ‘inverted pendulum problem’ Compliance Limitations of PID control COM2009-3009 Robotics: Lecture 6 slide 35 35 © 2023 The University of Sheffield Any Questions ? (or you can post on the ‘Lecture’ Discussion Forum) COM2009-3009 Robotics: Lecture 6 slide 36 36 18 © 2023 The University of Sheffield Next lecture … Reaching & Grasping COM2009-3009 Robotics: Lecture 6 slide 37 37 19