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Auxiliary controller design for low-cost rigid manipulator with built-in controller
Introduction
From a commercial and marketing point of view, “personal robots” must be cheap and user-friendly. A common practice for the current low-cost manipulators in the market is to adopt “all-in-one” actuators, e.g., Dynamixel AX-12A actuators from ROBOTIS. These actuators are integrated with DC motor, sensors, transmission system and position control unit, provide high load-to-weight ratio and relieve the practitioners from designing reduction system and control unit.
However, constrained by the budget, some low-cost manipulators adopt limited-performance all-in-one actuators, for which the built-in controllers are usually simple independent joint controllers, such as PID, PD or even P controller (if no velocity sensor or observer is avail-able). Since the coupling dynamic effects between joints are ignored [13], this leads to limited performance and control precision of the manipulator [100].
The most used closed-loop controller of a servo system is PD (proportional-derivative) controller [31]. Linear analysis suggests that with high gains, precise tracking performance can be achieved. However, high gains drive easily actuators into saturation, besides, for low-cost actuators, reliable velocity feedback with high frequency is hard to obtain. That is why some commercial smart actuators (e.g., Dynamixel AX-12 and AX-18 series) use only P controller, in this case, undesirable static error cannot be avoid when a large load on the motor axis is exerted.
To improve the control performance for position controlled robot manipulators, many works have been done to provide them with torque control capability, since torque controlled robots are preferred to achieve high performance [78]. Approach to use torque sensors, such as strain gauge [33] or optical torque sensor [34] techniques, is proposed in [35], this joint torque sensing method is adopted in [54]. However, since strain gauges are sensitive to ambient temperature variations, varying offset are often generated. For optical torque sensors, artificial flexibility inside a joint usually needs to be performed to convert the joint torque to joint deformation, while introducing joint flexibility may cause undesired performance such as vibration. In [41], a torque-position transformer for position controlled robots is introduced, with which desired joint torque is converted into instantaneous increments of joint position inputs, this approach has been implemented on the Honda ASIMO robot arm. However, this transformer is based on the total knowledge of the built-in controller, which, designed by the manufacturer of the actuators, is usually unknown or partly unknown to individual users. Besides, to calculate the desired torque, the model of the whole manipulator is indispensable, if it is not available, identification of the whole system model should be considered.
In this chapter, we consider the set-point regulation for a low-cost manipulator equipped with built-in controller. The objective is to design an auxiliary controller, without any additional sensors, to reduce the steady-state error which could not be handled by the built-in controller.
The remainder of this chapter is organized as follows: the statement and analysis of the control problem for low-cost rigid position-controlled manipulators are introduced in Section 2.2. In Section 2.3, dynamics model and model parameter identification method are given. Section 2.4 details the adaptive controller design and its implementation on discrete systems. Simulation results are presented in Section 2.5 with comparison to an auxiliary PID controller. Finally, the conclusion is given in Section 2.6.
Problem statement and analysis
Problem statement
Consider a n-DOF revolute robot manipulator, each joint is driven by an all-in-one actuator, each actuator is embedded with a controller.
For each joint, Figure 2.1 shows the system under the built-in controller. u is the input for the system, which could be the desired angular position. x denotes the joint position and serves as the feedback, corrupted with measurement noise. τm is the torque working on the motor, controlled by the built-in controller, based on the position error e, where e = u −x.
The following assumptions are made:
τm = P(u −x) −Dx˙ + ν, (2.1)
where P > 0 is the proportional gain, D ≥ 0 is the derivative gain, while ν denotes the unmodeled error and is bounded.
Assumption 2.2. The manipulator is rigid and time delay is not considered.
Assumption 2.3. The manipulator is of light weight and of small size.
Assumption 2.4. The gear ratio of the manipulator is rather small (e.g., less than 1:200).
Assumption 2.5. The angular velocity and acceleration of the manipulator are limited.
Assumption 2.6. Parameters of the dynamic model of the manipulator, e.g. joint mass, initial moments, first moments, etc, are unknown.
Assumption 2.7. Only joint angular position feedback is available for the user and is corrupted with noise.
The objective is to reduce the steady-state error which could not be done by the built-in controller, without any additional sensors.
Problem analysis
Since the built-in controller presents steady state error with u = xd , where u denotes the input for any joint, and xd is the desired position for this joint, to reduce the position error, the way is to design an auxiliary controller u based on xd and position feedback x (and the derivatives of x if necessary), u will afterwards serve as the input for the previous joint plus controller system, the idea is illustrated in Figure 2.2.
Most of the model-based controller design methods are based on the centralized inverse dynamic model (1.3) of manipulators, which relates the input torque vector to the state variable vectors. However, in our case, as we have neither direct or indirect torque input (the torque to be exerted on the motor is decided by the built-in controller of which the model is unknown), nor torque measurements, centralized control seems impossible, as a result, we resort to independent joint control.
The most used independent joint controller is the PID controller. However, from a linear control theory point of view, PID controller with constant gains may not be suitable for a nonlinear and time-varying system. To seek a better performance, a model-based independent joint controller could be a better choice.
Modelling and identification
Dynamics modelling
Generally, the inverse dynamics model of an n-DOF robot arm with viscous friction can be denoted as: τ = M(q)q¨ +C(q, q˙)q˙ + F(q˙) + G(q), (2.2)
where q is the joint variable n-vector and τ is the n-vector of the forces or torques. M(q) is the inertia matrix, C(q, q˙)q˙ is the Coriolis/centripetal n-vector, F(q˙) is friction vector, and G(q) is the gravity vector. One of the properties of (2.2) is that M(q), C(q, q˙), F(q˙), G(q), τ are bounded, as summarized in Section 1.2.2.
Most robot manipulators are driven by servo motors. The motors are connected to the manipulator links through gear trains, where the robot dynamics appear as dynamic load, as illustrated in Figure 2.3. The dynamics of a DC motor can be represented by a second-order linear differential equation. For the ith joint, the dynamics of the motor can be written as [75]
Table of contents :
1 Introduction
1.1 Background and motivation
1.2 Previous work
1.2.1 Robot Dynamics
1.2.2 Robot Dynamics Properties
1.2.3 Control of rigid manipulators
1.2.4 Control of flexible-joint manipulators
1.3 Contribution of the thesis
1.4 Outline of the thesis
2 Auxiliary controller design for low-cost rigid manipulator with built-in controller
2.1 Introduction
2.2 Problem statement and analysis
2.2.1 Problem statement
2.2.2 Problem analysis
2.3 Modelling and identification
2.3.1 Dynamics modelling
2.3.2 Model Identification
2.3.3 Derivative estimation
2.4 Auxiliary adaptive controller design
2.4.1 Adaptive controller design
2.4.2 Numerical implementation of the adaptive controller
2.5 Simulation
2.5.1 Model description
2.5.2 Constant input
2.5.3 Model parameter identification
2.5.4 Auxiliary adaptive controller
2.5.5 Auxiliary PID controller
2.6 Conclusion
3 Identification and control of single-link flexible-joint robots using velocity measurement
3.1 Introduction
3.2 Problem statement
3.3 System model
3.4 Parameter identification using angular velocity measurement
3.5 PD controller with gravity compensation
3.5.1 Without parameter uncertainty
3.5.2 With parameter uncertainty
3.6 Two-stage adaptive controller using inertial sensors
3.6.1 Motor position reference design
3.6.2 Adaptive input design
3.7 Simulation
3.7.1 Model specification
3.7.2 Identification results
3.7.3 Controller performance
3.8 Conclusion
4 Identification and control using measurements of higher order derivatives
4.1 Introduction
4.2 Identification and control for single-link flexible-joint robots with acceleration measurement
4.2.1 Identification
4.2.2 Two-stage adaptive controller using accelerometer
4.2.3 Simulations for single-link flexible-joint robots
4.3 Generalisation to linear system with high-order derivative measurement
4.3.1 Preliminaries
4.3.2 Problem statement
4.3.3 Identification
4.3.4 State space representation for control design
4.3.5 Adaptive control
4.3.6 Robust control
4.3.7 Simulations for linear system
4.4 Conclusion
5 Experiment results
5.1 Introduction
5.2 Introduction of the experiment plate-form
5.2.1 Description of the robot arm
5.2.2 All-in-one actuator
5.2.3 Built-in controller
5.2.4 Propositions and physical constraints
5.3 Rigid case
5.3.1 Built-in controller performance
5.3.2 Model identification
5.3.3 Control strategy
5.3.4 Performance with auxiliary controller
5.4 Flexible-joint case
5.4.1 Inertial sensors
5.4.2 Built-in controller performance
5.4.3 Model identification
5.4.4 Controller performance
5.5 Conclusion
Conclusions and Perspectives
References
Appendix A Dynamic of a 2DOF Rigid Robot Manipulator
Résumé en français
List of publications
