Design of a few backstepping sliding mode based robust control techniques for robot manipulators

The basic methodology used in the thesis is the backstep-based sliding mode controller. IADSC Integral Adaptive Dynamic Surface Controller IBSMC Integral Backstep Sliding Mode Controller MAE Mean Absolute Error.

Robot manipulator

Although from the position control point of view the kinematic control is sufficient, however, the following disadvantages are faced during the implementation of the kinematic control method:. i) The load dynamics of the manipulator can affect the joint movement which can increase the steady state error if only kinematic control is implemented. ii). In practice, actuators in the manipulator joints have torque or force limits that are not taken into account by the kinematics. iii) Dynamic disturbances such as friction force cones, center of pressure positions [11] are also not taken into account in the kinematics. iv).

Literature review: Robust controllers for robot manipulators

Following the methodology of [48], these controllers for robot manipulators were developed based on the knowledge of the uncertainty limit. Another drawback of SMC is that its robustness is only guaranteed when the system states are on a sliding surface, and it is not immune to uncertainty in the reaching phase.

Motivation

The second important criterion is the structural simplicity of the controller and the minimum information requirement. In [117], a three-part torque control law was formulated, which required the estimation of joint torques based on data from the end-effector torque sensor and the robot model.

Contributions of the thesis

Proportional, integral, and derivative gains of the sliding surface are derived by inverse stepping. The structure of manipulator dynamics becomes more complex as the number of DoFs increases.

Organization of the thesis

In the second part of the chapter, the ABSMC-PID is designed for impedance control of robot manipulators as they interact with the external environment. Experimental validation of the proposed conversion methods as well as the proposed control laws, ABSMC-PID and ABFTSMC, are also presented in this chapter.

Motivation

In this chapter, an integral backward sliding mode controller (IBSMC) is proposed for dynamic control of nonlinear robot manipulator systems, and the block diagram of the proposed IBSMC is shown in Fig. IBSMC derives a sliding surface based on the posterior integral method and finally produces a discontinuous function as the derivative of the control input.

Figure 2.1: Block diagram: Integral Backstepping Sliding Mode Control

IBSMC design for robot manipulators

Without the control input to the augmented system (2.5), the input τ to the manipulator is obtained as follows: However, the cost of accuracy reduction is lower compared to the smoothness obtained from the control input of the IBSMC.

Integral Adaptive Dynamic Surface Controller

Since the relative degree of the system is increased by one, as shown in (2.5), the actual control input is now u= ˙τ. Simulation results demonstrate the suitability of the proposed IADSC in robot trajectory tracking tasks. Following are the issues that may arise in controller design: i) Controller gain design is an issue because it is generally done heuristically, normally selecting a very high gain value.

Table 2.6: Performance comparison for stabilizing task of 2 DoF manipulator

Summary

The controller is chatter-free and has the ability to resolve uncertainty without having to resort to observers. An adaptively tuned controller gain helps maintain the robustness of the controller against unknown and variable uncertainties and also helps in reducing unnecessary input power consumption during steady state. In addition, the controller gain is now adaptively adjusted to handle uncertainty with unknown bounds.

Motivation

Adaptive Backstepping Sliding Mode Controller with PID Slid- ing Surface

The integral of the error is considered as the first variable and is given as. Now s=0 is the sliding surface for the system and the introduction of the sliding variable changes (3.19) in the following: 3.21). The control law is now derived in two parts: (a) the equivalent control ueq and (b) the switching control usw. i) Stability of the adaptive law.

Figure 3.1: Simulation results: Tracking errors for 2DoF manipulator with Yang et al. ’s controller [2], IADSC and the proposed ABSMC-PID in presence of measurement noise

ABSMC-PID for hybrid impedance control of robot manipula- tors

The following Lyapunov function Vk is chosen for the stability analysis of the sliding surface and the adaptive law,. The interaction force will be zero when there is no contact of the end effector with the external environment. The movement of the end effector in the x-y plane and the y-z plane is shown in Figure 3.6 for both varying and constant c2 values.

Figure 3.5: Input torques for the manipulator joints with varying and constant impedance

Summary

Desired Variable impedance Constant impedance Vertical wall at x=0.12m. a) Movement of the end effector in the x-y plane. The norms of the input moments (||u||) for each term and their total variations (TV) are listed in Table 3.3, and it is observed that these are comparable for both varying and constant c2 values. As a solution, a time-delay-based estimation of the manipulator model is proposed in the next chapter, which will lead to a model-free controller design.

Introduction

Controller Design

In this section, the stability of the controlled system based on Lyapunov and the adaptive law is analyzed. In order to prove the stability of the controlled system, we first establish the boundedness of the time lag estimation error. Furthermore, using property 3, it was found that the inertia matrix Mh of the manipulator is bounded.

Simulation Results

Moreover, the control input produced by the proposed method chatted much lower than the RFTSC by Zhao et al.[3], as evident in Figure 4.2. The performance of the proposed controller is then investigated for a continuous time trajectory and the results are compared with the RFTSC controller by Zhao et al. The proposed controller uses almost the same amount of control energy as the controller by Zhao et al.

Figure 4.1: Tracking response with the proposed controller and RFTSC proposed by Zhao et al

Summary

However, the total variation (TV) measurements of both controllers in Table 4.1 show that the proposed ABSMC has less jitter than the RFTSC. The tracking results in Table 4.1 clearly show that the proposed controller can maintain good tracking performance, while Zhaoet al. The fast sliding mode of the terminal, combined with the backward step and adaptively set controller gain, gives the controlled system robustness despite the likely error of the modeling estimate due to TDE.

Introduction

All such uncertainties that are not taken into account by the internal controller in the servo motors will be addressed by the dynamic controllers proposed in the previous chapters, mainly the ABSMC-PID and the ABFTSMC. The chapter is structured as follows: section 5.2 describes the coordinated left (COOL) robot arm and its parameters. Experimental results using the ABSMC-PID and the ABFTSMC proposed in the thesis are presented in Section 5.5 and Section 5.6 to validate the proposed torque-to-position converter and to study the effects of including the dynamic controller in the loop for a commanded position. robot manipulator.

Position controlled manipulator: The Coordinated Links (COOL) robot arm

195] proposed a joint level controller for a decentralized manipulator system containing Dynamixel AX series servo and designed the controller considering proportional (P) action in the internal control of the servo. However, adopting a simplification strategy and using only nominal motor parameters cannot mitigate structured and unstructured uncertainties that are present in the system and affect the motor dynamics. Communication between the computer and the robotic arm is performed via the communication device called the USB2Dynamixel [197].

Joint actuators

A linear potentiometer mechanically connected to the transmission output shaft serves as a position sensor. The magnitude and polarity of the motor's armature voltage controls the servo's angular velocity and direction. The block diagram presentation of Dynamixel RX-28 and RX-64 [5] servomotors is shown in Figure 5.3.

Figure 5.2: The experimental set-up for the robot arm

Torque to position converter

Based on the position feedback, the microcontroller in the servo outputs control using the PID controller programmed in it to move the servo to the desired common position sensed by the transmitter [5]. The torque-to-position converter will produce a position command qcmd as shown in Figure 5.5 based on the resulting torque τ of the dynamic controller obtained according to the desired angular position of the joint qd, the desired angular velocity of the joint ˙qd and the angular acceleration of desired union ¨qd. This conversion can be used to implement an external dynamic control circuit on the robot arm as shown in Figure 5.5.

Experimental results with ABSMC-PID

This study is conducted to test the reliability of the proposed ABSMC against the built-in P controller. The trajectory tracking performance with the direct position command, ABSMC-PID, and the ABSMC-NPID is compared in Figure 5.6, and the tracking performance of all three methods is compared in Table 5.5. Secondly, from the error responses in Figure 5.6, it can be noted that in the case of coupled movements with load, the performance of the built-in P-type controller is not consistent (seen from the tracking results of connection 2), while the dynamic controller ensures consistent performance in all three joints.

Figure 5.5: Block diagram of proposed dynamical control

Experimental results with ABFTSMC

The inclusion of the nonlinear PID sliding surface is able to further improve the performance of the proposed ABSMC by reducing peak overshoot as can be observed in Figure 5.6 and Table 5.5. From Figure 5.7 and Figure 5.8, it is clear that in the case of position-controlled robots, instead of directly applying the desired trajectory as the command input to the system, if the dynamic controller is used first, better tracking results are achieved. Moreover with the proposed ABFTSMC, the overshoot is also lower than with ABSMC-NPID.

Figure 5.8: Results with direct position command, proposed ABMSC-NPID and ABFTSMC

Summary

Conclusions

The thesis attempted to implement the proposed dynamic control methods in digital servo systems with position command. A torque-to-position command conversion method was used to convert the generated torque profile into a position command for driving servo motors in a robot manipulator. Experimental studies have shown promising potential for practical applicability of the proposed method to position-controlled robots.

Scope for Future Work

Dynamic modeling of rigid manipulators

Characteristics of symmetric positive definite block matrix using Schur’s complement

Dynamics of the cart-pendulum system used in (A.13)

Derivation of IBSMC for cart-pendulum system

It can be observed from above that solution z1 will still not be driven to the equilibrium point, i.e. the origin. A new Lyapunov function is now defined in terms of the two regulatory variables rz1 and z2 as V2=V1+1. The derivative of the Lyapunov function V2 will have the following expression with η2v as the control input.

Coupled SMC proposed by Park and Chwa [1] for stabilization control of cart-pendulum system

From the above equation, it can be observed that on the horizontal plane, i.e. when q2 = π2, the system has a singularity in its zero dynamics. So when the pendulum crosses the horizontal plane, it can lead to unlimited control input. But while doing so, it should also be noted that the control law is able to pump enough energy into the system for it to cross the horizontal plane.

Proof of Lemma 4

Model of 2DoF manipulator used in Yang et al. [2]

In the simulations it is assumed that only the position feedback is available and it has an additional uniform noise with the limits ±0.00001 rad.

Disturbance observer based adaptive robust controller proposed by Yang et al. [2]

For any local system, ˆwi is the compensation term used by the disturbance observer for the disturbance term wi, where.

Dynamics of the 3DoF manipulator simulated in the Coordi- nated Links (COOL) robot arm

Proof of Theorem 6

Time derivative of the sliding manifold used in Chapter 4

RFTSM controller by Zhao et al. [3] used in Chapter 4

Brezak, "Passive finite-dimensional repetitive control of robot manipulators," IEEE Transactions on Control Systems Technology, vol. Wu, "A robust MIMO terminal sliding mode control scheme for rigid robot manipulators," IEEE Transactions on Automatic Control, vol. Singh, "Adaptive model after control of nonlinear robotic systems,” IEEE Transactions on Automatic Control , vol.