Steps in Development Control Systems for Robots

Steps in Development Control Systems for Robots Modelling Robot Behaviors Kinematics Dynamics Parameters Estimation c Anton Shiriaev. 5EL158: Lecture 9 p. 1/21

Steps in Development Control Systems for Robots Modelling Robot Behaviors Kinematics Dynamics Parameters Estimation Choice of Actuators and Gears c Anton Shiriaev. 5EL158: Lecture 9 p. 1/21

Steps in Development Control Systems for Robots Modelling Robot Behaviors Kinematics Dynamics Parameters Estimation Choice of Actuators and Gears Choice of Sensors and Their Allocation c Anton Shiriaev. 5EL158: Lecture 9 p. 1/21

Steps in Development Control Systems for Robots Modelling Robot Behaviors Kinematics Dynamics Parameters Estimation Choice of Actuators and Gears Choice of Sensors and Their Allocation Choice of Control Architecture Linear vs. Nonlinear Compensation for Friction, Delays, Lack of Velocity Measurements Anti-Windup Methods Adaptive Mechanisms for On-line Parameters Estimation Robustness... c Anton Shiriaev. 5EL158: Lecture 9 p. 1/21

Lecture 9: Independent Joint Control Conceptual Control Loop for One Joint c Anton Shiriaev. 5EL158: Lecture 9 p. 2/21

Lecture 9: Independent Joint Control Conceptual Control Loop for One Joint Actuator Dynamics c Anton Shiriaev. 5EL158: Lecture 9 p. 2/21

Lecture 9: Independent Joint Control Conceptual Control Loop for One Joint Actuator Dynamics Dynamical Model of a Robot with One Joint c Anton Shiriaev. 5EL158: Lecture 9 p. 2/21

Lecture 9: Independent Joint Control Conceptual Control Loop for One Joint Actuator Dynamics Dynamical Model of a Robot with One Joint Controller Design PD and PID Tuning; Feedforward Control; Problems with Gear-Boxes; State Space and Observer Based Controllers c Anton Shiriaev. 5EL158: Lecture 9 p. 2/21

Basic structure of a feedback control system. It is often appropriate for controlling robots. c Anton Shiriaev. 5EL158: Lecture 9 p. 3/21

Principle of operation of a permanent magnet DC motor: A current-carrying conductor in magnetic field experience a force F = i φ here i is is the current; φ is the magnetic flux c Anton Shiriaev. 5EL158: Lecture 9 p. 5/21

Principle of operation of a permanent magnet DC motor: A current-carrying conductor in magnetic field experience a force F = i φ ( τ m = K i φ ) i here i is is the current; φ is the magnetic flux c Anton Shiriaev. 5EL158: Lecture 9 p. 5/21

Principle of operation of a permanent magnet DC motor: Whenever a conductor moves in a magnetic field, the voltage V b is induced and it is proportional to a velocity of the conductor ( ) V b = K φ d dt θ m c Anton Shiriaev. 5EL158: Lecture 9 p. 5/21

Relations between the armature current, voltage,rotor velocity and motor torque are L d dt i + Ri = V V b ( ) V b = K φ d dt θ m ( τ m = K 1 i φ ) i = K m i c Anton Shiriaev. 5EL158: Lecture 9 p. 5/21

Dynamical Model of a Robot with One Joint Lumped model of a single link with actuator/gear transmission. c Anton Shiriaev. 5EL158: Lecture 9 p. 7/21

Dynamical Model of a Robot with One Joint Lumped model of a single link with actuator/gear transmission. In terms of the motor angle θ m the equation of motion is ( ) J d 2 m θ dt 2 m = τ m 1 ( ) r τ d l B m dt θ m = K m i 1 r τ l B m ( d dt θ m ) c Anton Shiriaev. 5EL158: Lecture 9 p. 7/21

Dynamical Model of a Robot with One Joint Augmenting the mechanical and electrical models, we obtain: ( ) ( ) L d dt i + R i = V K d b dt θ m ( ) ( ) J d 2 m θ dt 2 m + B d m dt θ m = K m i 1 r τ l c Anton Shiriaev. 5EL158: Lecture 9 p. 8/21

Dynamical Model of a Robot with One Joint Augmenting the mechanical and electrical models, we obtain: ( ) ( ) L d dt i + R i = V K d b dt θ m ( ) ( ) J d 2 m θ dt 2 m + B d m dt θ m = K m i 1 r τ l System is linear! We can use classical methods for controlling it! c Anton Shiriaev. 5EL158: Lecture 9 p. 8/21

Dynamical Model of a Robot with One Joint The transfer function from V (s) to Θ m (s) is G {v θ} (s) = Θ m(s) V (s) = K m [ (Ls + R)(Jm s + B m ) + K m K b ] s c Anton Shiriaev. 5EL158: Lecture 9 p. 9/21

Dynamical Model of a Robot with One Joint The transfer function from V (s) to Θ m (s) is G {v θ} (s) = Θ m(s) V (s) = K m [ (Ls + R)(Jm s + B m ) + K m K b ] s The transfer function from τ l (s) to Θ m (s) is G {τl θ}(s) = Θ m(s) τ l (s) = 1 r (Ls + R) [ (Ls + R)(Jm s + B m ) + K m K b ] s c Anton Shiriaev. 5EL158: Lecture 9 p. 9/21

Dynamical Model of a Robot with One Joint The transfer function from V (s) to Θ m (s) is G {v θ} (s) = Θ m(s) V (s) = K m [ (Ls + R)(Jm s + B m ) + K m K b ] s The transfer function from τ l (s) to Θ m (s) is G {τl θ}(s) = Θ m(s) τ l (s) = 1 r (Ls + R) [ (Ls + R)(Jm s + B m ) + K m K b ] s If L R 0 G {v θ}(s) K m [ R(Jm s + B m ) + K m K b ] s c Anton Shiriaev. 5EL158: Lecture 9 p. 9/21

Dynamical Model of a Robot with One Joint The transfer function from V (s) to Θ m (s) is G {v θ} (s) = Θ m(s) V (s) = K m [ (Ls + R)(Jm s + B m ) + K m K b ] s The transfer function from τ l (s) to Θ m (s) is G {τl θ}(s) = Θ m(s) τ l (s) = 1 r (Ls + R) [ (Ls + R)(Jm s + B m ) + K m K b ] s If L R 0 G {v θ}(s) K m [ R(Jm s + B m ) + K m K b ] s Correctness of reduction of the model based on size of the parameter can be justified via e.g. balanced model reduction c Anton Shiriaev. 5EL158: Lecture 9 p. 9/21

Example: Suppose that K b = 1 and G {v θ} (s) K m [ R(Jm s + B m ) + K m K b ] s = 1 ( s + 1 ) s Design a PD-controller that closed loop poles are at 3, 1 c Anton Shiriaev. 5EL158: Lecture 9 p. 11/21

Example: Suppose that K b = 1 and G {v θ} (s) K m [ R(Jm s + B m ) + K m K b ] s = 1 Js 2 + Bs Find a PID-controller gains such that that closed loop is stable c Anton Shiriaev. 5EL158: Lecture 9 p. 12/21

Dealing with Input Saturation The closed loop system can be drastically different if we take into account signals limits in the loop! c Anton Shiriaev. 5EL158: Lecture 9 p. 13/21

Idea of Feedforward Control Transfer function F(s) is new parameter for design c Anton Shiriaev. 5EL158: Lecture 9 p. 15/21

Idea of Feedforward Control Transfer function F(s) is new parameter for design Not any F(s) can be used: F(s) should be stable; F(s) should be proper c Anton Shiriaev. 5EL158: Lecture 9 p. 15/21

Idea of Feedforward Control Transfer function F(s) is new parameter for design Not any F(s) can be used: F(s) should be stable; F(s) should be proper Reasonable choice is F(s) 1/G(s) c Anton Shiriaev. 5EL158: Lecture 9 p. 15/21

Gear-Boxes Lossless transmission means that τ m θm = τ s θs ; c Anton Shiriaev. 5EL158: Lecture 9 p. 17/21

Gear-Boxes Lossless transmission means that τ m θm = τ s θs ; Gear-box always introduces additional friction and delay; c Anton Shiriaev. 5EL158: Lecture 9 p. 17/21

Gear-Boxes Lossless transmission means that τ m θm = τ s θs ; Gear-box always introduces additional friction and delay; Gear-box can complicate the dynamics and limit the performance. c Anton Shiriaev. 5EL158: Lecture 9 p. 17/21

Flexibility of in Harmonic Drive Transmission The dynamics are J l θ l + B l θ l + k ( ) θ l θ m = 0 J m θ m + B m θ m k ( ) θ l θ m = u where u is a control torque. c Anton Shiriaev. 5EL158: Lecture 9 p. 18/21

Flexibility of in Harmonic Drive Transmission The dynamics are J l θ l + B l θ l + kθ l J m θm + B m θm + kθ m = kθ m = kθ l + u where u is a control torque. c Anton Shiriaev. 5EL158: Lecture 9 p. 18/21

Flexibility of in Harmonic Drive Transmission The dynamics are k Θ l (s) = p l (s) Θ m(s) = Θ m (s) = 1 p m (s) k J l s 2 + B l s + k Θ m(s) ( kθl (s) + U(s) ) = kθ l(s) + U(s) J m s 2 + B m s + k c Anton Shiriaev. 5EL158: Lecture 9 p. 18/21

Flexibility of in Harmonic Drive Transmission The dynamics are k Θ l (s) = p l (s)p m (s) k 2U(s) = k J m J l s 4 +(J l B m +J m B l )s 3 +(k(j m +J l )+B m B l )s 2 +k(b m +B l )s U(s) c Anton Shiriaev. 5EL158: Lecture 9 p. 18/21

Flexibility of in Harmonic Drive Transmission The dynamics are k Θ l (s) = p l (s)p m (s) k 2U(s) = k J m J l s 4 +(J l B m +J m B l )s 3 +(k(j m +J l )+B m B l )s 2 +k(b m +B l )s U(s) k J m J l s 4 +k(j m +J l )s 2 U(s) c Anton Shiriaev. 5EL158: Lecture 9 p. 18/21

PD controller with θ m or θ l feedbacks c Anton Shiriaev. 5EL158: Lecture 9 p. 19/21

The root-locus of the closed loop system with θ m -feedback and PD = K P + K D s = K(a + s) c Anton Shiriaev. 5EL158: Lecture 9 p. 20/21

The root-locus of the closed loop system with θ l -feedback and PD = K P + K D s = K(a + s) c Anton Shiriaev. 5EL158: Lecture 9 p. 21/21