ROBOTICS Control of Robot Manipulators TEMPUS IV Project

ROBOTICS Control of Robot Manipulators TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS

Tasks Control Robot Level task Pe(t) Trajectory Planning (IK, J, etc) controller Power electronics Current to motors 2

Tasks Control Robot Level task Pe(t) Trajectory Planning (IK, J, etc) controller Power electronics Current to motors 3

Control • 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” • Dynamics Based control – Use dynamics model to facilitate control • • Compute torque feedforward Inverse Dynamic Control Operation Space control And Compliance, Impedance, Force…. TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 4

Jointed system components TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 5

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) TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 6

Block Diagram of PE controller for a single joint Reference angle Energy source (current) control signal torque Error signal PE controller u = Kpe e u measured joint position, qm TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS Motor +reduction +transmission Joint position sensor Joint Actual joint position, q ROBOTICS 7

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: TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 8

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) Applied current, amp Motor torque, Nm Motor constant, Nm/amp TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 9

actuator current vs torque TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 10

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) disturbance control input Rotational inertia of joint, kg m^2 Effective damping (friction, back emf), Nm/amp TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 11

Independent Joint Control • Control each joint independently without “communication” between actuators • Basic Steps: ü Model actuator ü Use kinematics to obtain setpoints for each joint (recall trajectory planning – chapter 5) – Develop a controller for each joint • Error for joint i: TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 12

Proportional control for each joint • Input proportional to position error: • Neglect disturbance, wlog set reference position to zero • or TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 13

Proportional control for each joint • Second order linear differential equation: • has general form solution: – where TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 14

Block Diagram of PE controller + - 1 s(Js+F) Kpe Sensor transfer function TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 15

Three solutions What does this term do? TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 16

Three solutions • Over-damped (w 2 > 0) • Critically damped (w 2 = 0) 17

Three solutions • Under-damped (w 2 < 0) – w has complex roots – Oscillates with frequency If B is small and KPE is large: unstable! 18

Example Step Responses (1 radian) 19

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) TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 20

Block Diagram of PE controller + - 1 s(Js+F) Kpe Ki(1/s) Kd(s) Sensor transfer function TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 21

Set Gains for PID Controller • wlog set (we already have ) • Convert to third order equation • Solution will be of the form – where TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 22

Set Gains for PID Controller • Critically damped when w = 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 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 23

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 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 24

Dynamics: Equations of Motion • Dynamic Model – forces/torques motion of manipulator+load • Equations of Motion • Ideally we can use – Modeling errors – Friction – synchronization TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 25

Literature: • Spong, Hutchinson, Vidyasagar: “Robot Modeling and Control”, Wiley, 2006 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 26