Robot Manipulator Control. Hesheng Wang Dept. of Automation

Transcription

1 Robot Manipulator Control Hesheng Wang Dept. of Automation

2 Introduction Industrial robots work based on the teaching/playback scheme Operators teach the task procedure to a robot he robot plays back eecute the motion Intelligent robots work without teaching Robots plan its motion automatically Robots eecute the motion planned o carry out tasks successfully, a robot needs to input proper inputs to their joint actuators to eecute the motion taught/planned.

3 Problem Definition Robot Control is a problem of designing a proper input to force a robot to carry out a task or a motion as precisely as possible. Robot control problems can be classified as follows according to the reuirements: Position control: Control a robot to a desired position rajectory control: Control a robot to follow a desired geometric path or a time-varying trajectory Force control: Control a robot to apply a desired force to the environment or other objects Hybrid position/force control: Control position and force of a robot.

4 Controller o control motion of a robot based on the information collected by the sensors and instructions. Motion instruction Controller Robot Sensor Feedback

5 Control Method Open-loop Control: Directly compute the motion input from the motion command without using feedback information. reference Controller robot -simple - big errors - no robustness

6 Control Method Closed-loop control: Control the robot based on the instruction and the sensor feedback. - High accuracy - Robust reference Controller Robot Sensor feedback

7 Review of Stability heory

8 Nonlinear State-Variables Systems he robot dynamics has the form: τ G C H, From dynamic euation: τ H G C H, y τ H G C H, State euation of robot dynamics y,, Let

9 Nonlinear State-Variables Systems In general, the state euation of nonlinear systems: f,u System input y h,u System output State variables vector

10 Positive-Definite Matri An n n matri H is positive - definite if n H, R and An nn matrih is positive- semidefinite if n H, R An n n matri H is negative - definite if n H, R and An nn matrih is negative- semidefinite if n H, R All eigenvalues of a positive-definite matri are positive. All eigenvalues of a positive-semidefinite matri are non-negative All eigenvalues of a negative-definite matri are negative. All eigenvalues of a negative-semidefinite matri are non-positive

11 Euilibrium Point Euilibrium point: A state e of a system in which the velocity of the system is eual to zero. System : Let f,t the euilibrium point satisfies f e,t Euilibrium point of a system may not be uniue Ball staying on the top Ball staying at the bottom

12 Lyapunov Stability X e is stable in the sense of Lyapunov at t, if starting close enough to e at t, the system will always stay close to e at later time. Definition : e is stable in the sense of Lyapunov at t if for any given, there eists a positive, t such that if t -, t, then t -, t t e e e System trajectory

13 Lyapunov Stability It should be noted that stability does not mean that the state of the system will go to the euilibrium point as time goes to the infinity. e Ball staying on the top Instable e Stable with or without friction

14 Asymptotic Stability Xe is asymptotically stable if it is both stable and convergent. i.e. there eists a positive such that if t e, then limt t e e

15 Global Eponential Stability eis globally eponentially stable if there eists and such that for, t - a e t e e a t - e Eponential function rajectory of the error t

16 Lyapunov heorem Given the nonlinear system: f, with an euilibrium point at the origin =. Let N be a neighborhood of the origin. Suppose that is within the region N. he origin is stable in the sense of Lyapunov if there eists a scalar function V such that V is positive-definite in region N V is negative-definite or negative-semidefinite in region N he origin is asymptotically stable if V is positive-definite in region N V is negative-definite in region N If the N is the entire state space, the stability is global; otherwise, the stability is local

17 LaSalle heorem Given the nonlinear system: f with an euilibrium point at the origin =. Suppose that there eists a scalar function V such that V is positive-definite, V t he system will be convergent to the maimum invariant set within the region defined by V t. If the maimum invariant set contains the origin only, the origin is asymptotically stable.

18 Stability Analysis Barbalat Lemma : If ft is uniformly continuous and if t lim f t dt a a, then lim t t f t Uniform continuity : A function ft is uniformly continuous if any, there eists such that for any t and t if t - t, then ft - ft for A sufficient condition for uniform continuity: A differentiable function ft is uniformly continuous if its derivative is bounded.

19 Position Control Position control is a problem of properly designing joint inputs of a robot manipulator so as to drive it to a desired configuration position and orientation he desired position is a constant vector he desired velocity and acceleration are zero. Position control can be classified into Position control in joint space Position control in Cartesian space.

20 Joint Servo System Reference Computer D/A Converter Amplifier Actuator Link Sensor Computer or MPU: carry out the necessary computation in a control law D/A converter: ransform digital output of the computer to analogical voltages Amplifier: Amplify the input voltage to the actuator. Actuator: Drive the link Sensors encoder, etc: Measure the joint angle and velocity.

21 Actuator Dynamics An actuator usually delivers torue to a robot link via a reduction mechanism. i s i Motor Motor shaft m, i link Motor torue: Reduction gears m H m s L Gear ratio: s orue consumed by the reduction mechanism L, i Load of the link

22 Dynamics of DC motor Consider the euivalent circuit of a DC motor m Ki 3 K : motor torue proportional constant Applying Kirchnoff s voltage law di L Kbi Ri V 4 dt K b : back electromotive force emf constant + R V L Neglecting the inductance and considering : K b s i Ri V 5 R: Armature resistance L: Armature inductance V: Armature voltage i

23 Robot Dynamics with Actuator Dynamics Combining and 5 leads to the relationship between motor input voltage and the torue delivered to the link: L Ks R V H m s KK R b i s Substituting this into the robot dynamics: H H B H S, m G u If s i are very large, H m B G If there is no gravity: H u m B u H u B m i diag Ks R H s diag i mi s i i Ks V, R KiK R bi V Linear system Kns,..., R n n V n

24 PD Feedback Control Control objective: Control the position of the robot to a desired position d. Control Law : u -A - B : B A : he closed-loop dynamics: H H d : S, G Position error Position gain matri Velocity gain matri he euilibrium point: Δe A G AΔ B At the euilibrium point, the position error is not zero. he error is called offset error

25 PD Feedback Control Stability of the euilibrium point H U t V Introduce the following candidate of Lyapunov function Gravitational Potential energy A H U t V G U Form the closed-loop dynamics and noting B t V Vt is a Lyapunov function of the closed-loop dynamics, so the euilibrium is stable. However, the position error does not go to zero he position goes to zero when there is no gravity force

26 Offset Error he PD controller leads to an offset error: Δ A G How to kill the offset error Design a robot such that the gravity term is zero Increase the gain- high-gain control Having a gear-ratio reduction mechanism leads to high gain control Compensate for the gravity force on-line

27 PD Feedback Plus Gravity Compensation Control Law : u -A - B G Gravity compensator he gravity force is calculated on-line so as to cancel its effect his controller was developed by Arimoto and akegaki, 98 he closed-loop dynamics: H H S, AΔ B he euilibrium point:, Δ

28 Convergence of Position Error Introduce the following candidate of Lyapunov function V t H Α V t H H A Form the closed-loop dynamics and noting V t S, B B From the closed - loop dynamics andvt, the invariant set with V t contains only,. herefore, LaSalle heorem means lim t, lim t,

29 PID Feedback Control Control Law : u -K - K K K K 3 K 3 t dt : position error gain matri P - gain : velocity gain matri D - gain : Integral gain matri I - gain By introducing I-feedback, it can kill the offset error in PD feedback control However, the PID controller only gives local asymptotic stability. he proof is very complicated

30 Position Control in Cartesian Space Why? Most tasks are specified in a Cartesian space. We need to position the end-effector to desired position and orientation Control in the joint space will lead to offset errors due to errors in robot kinematics models. Difference from joint space control Directly use feedback of position in Cartesian space.

31 Position Control in Cartesian Space - - Law : Control G B A J u Cartesian space position error in : Robot Jacobian : d J B A J S H H Δ, Α H t V J J constant vector is a and d, B S B Α J H H t V hen, From LaSalle heorem, we can further prove lim t Position of the end-effector Desired position he closed-loop dynamics Consider the positive-definite function: herefore, he system is stable.

32 rajectory racking Control rajectory tracking control is a problem of designing proper joint inputs for a robot to trace a pre-specified trajectory presented by d d d d t, d t and d t t : desired position t : desired velocity t : desired acceleration In trajectory tracking, the nonlinear inertia, centrifugal and Coriolis forces play important roles. heir compensation is one of the key issues in controller design

33 Computed orue Method Linearization of the nonlinear dynamics Change the nonlinear system to a linear system by state feedback. he idea is to use on-line measurement of position and velocity to compensate for the nonlinear forces. he robot dynamics H H S, G τ Design the joint input as follows: τ H S, G τ' Substituting it to the dynamics: H τ' A new input 3

34 Computed orue Method Design the new input as follows: τ' H u 4 Substituting this into e. 3 in the previous slide: u 5 New input his is a linear system. It is easy to design the following controller: u t d t K t K t 6 where t t t, t t t he closed-loop system is t K t K t d d d 7 7 Is a second order system. It is asymptotically stable if K and K and positive-definite. hen from, 3 and 5, joint input is τ H S, G H d t K t K t 8 d

35 Controller: τ Computed orue Method H S, G H d t K t K t It uses real trajectory of a robot manipulator to compensate for its nonlinear forces centrifugal and Coriolis forces on-line. he on-line compensation reuires an efficient algorithm to solve the inverse dynamics in real-time. he recursive Newton-Euler formulation can solve the inverse dynamics in real-time. he measurement errors of joint positions and velocities will greatly affect the control performance