Abstract
Abstract-This paper presents the low level control of an holonomic robot with four omnidirectional wheels. A robust control technique named Quantitative Feedback Theory (QFT), based on an uncertain linear model has been selected to design the PID speed controllers for the four-wheeled robot. A piecewise model has been estimated by means of the least squares estimation approach based on experimental results of the robot in closed loop. In particular, the control is designed using this piecewise model. The performances of the proposed approach are analyzed in real time domain.
I. INTRODUCTION
Robots with omnidirectional locomotion have the ability to move in any direction and rotate simultaneously, fact that gives them great freedom of movement and allows them to maneuver in tight and complex spaces. Omnidirectional mobile robots are a kind of holonomic robots. They have better maneuverability compared to non-holonomic robots. This capability of movement is achieved by providing the wheels for mobility in lateral displacement. Today omnidirectional robots are very useful in industrial and home applications as the popular vacuum cleaner robots. In particular, in this work an omnidirectional holonomic mobile robot of four wheels is used.
I. INTRODUCTION
Different techniques have been applied in order to solve the control problem for omnidirectional mobile robot. Modelpredictive control (MPC) techniques are commonly applied to trajectory following [1] . The application of MPC requires a precise model of the system which, in practice, is difficult to achieve due to nonlinearities present in the physical systems. The nonlinearities in the mobile robots are mainly due to the friction. Many authors have addressed the problem of friction compensation, using model based and model free-based techniques [2] . The friction model-based approach consists of feedforward and feedback control that introduced the friction model into the system to cancel out the nonlinear effect. These techniques require a complete model of the friction behavior or adaptive algorithms for estimating the friction parameters online. The friction model-free approach consists in applying linear or nonlinear control for compensating the friction and improves the overall performance of the system. In general, these techniques are based on modifications of PID controllers in order to cope with the nonlinear effects of friction.
I. INTRODUCTION
In this paper the model-free approach based on PID structure will be applied to compensate the nonlinearities due to the friction of an omnidirectional robot. More specifically, the cascade control strategy, shown in Fig. 1 , will be applied. It consists of two loops: the high level loop (or outer-loop controller) that intends to adjust the control position and orientation to follow the commanded trajectory and a low level loop (or inner-loop) that follows the body rate command given by the high level loop. Both control loops are interconnected by the transformations of direct and inverse kinematics of the robot itself. Thus, the controller output paths setpoints results in four-wheel speeds through the matrix inverse kinematics of the robot and otherwise feedback sensors (encoders) that provides movement of the four wheels is translated into the position and orientation of the robot dynamics by direct kinematics matrix. High-level control with strategies for trajectory planning is discussed in [3] - [4] - [5] . In this paper we focus on the design of a robust low-level controller or speed control that controls the speed of the four wheels, with the objective to compensate nonlinearities.
I. INTRODUCTION
This work presents the design of a control strategy taking into account the uncertainties and nonlinearities of an omnidirectional mobile robot. This technique of robust control, named Quantitative Feedback Theory (QFT) uses linear interval models.
I. INTRODUCTION
The paper is organized as follows: in Section II the omnidirectional robot platform is described. Section III shows the method used for estimating the robot model. Section IV describes QFT technology and the design of QFT controller. Section V shows the experimental results. Finally, the conclusions can be found in Section VI.
II. ROBOTIC PLATFORM DESCRIPTION

A. Physical description
The Robotic platform is a four-wheel holonomic mobile robot developed by ST presenting a great maneuverability and effectiveness. The omnidirectional feature is reached thanks to the characteristics of the wheels, which roll forward like normal wheels but can slide sideways at the same time, allowing almost independent tangential, normal and angular velocities (holonomic property).
A. Physical description
It consists of a round structure of 25 cm of diameter supported by four omnidirectional wheels (see Fig. 2 ). With all components mounted its total weight is about 4.3kg. The robotic platform is equipped with four DC motors, one DC-DC converter, two motor drives board, eight ultrasonic ranges and a 12V deep cycle lead acid battery of 1.2 Ah. Each motor is equipped with incremental encoder counting N = 3600 pulses/turn.
B. Control System Architecture
The platform has two-level control architecture. The main control is conducted by an onboard computer connected via USB communication to the microcontroller on the robot (see Fig. 3 ).
B. Control System Architecture
The steering control module is run in a Net-PC (a Cortex-M4 with CPU working at 168MHz with 192KB of RAM) with Linux. The control algorithms are written in MATLAB and run with a sample time of 40 ms. Three working modes have been implemented: a) speed control for each wheel; b) current control of each motor; and, c) open loop control that allows sending individual PWM set point for each motor. Regarding the high-level algorithms navigation, external communications, algorithms fault tolerance, among others, can be implemented as tasks.
B. Control System Architecture
The low-level control layer is in charge of the execution of the high-level velocity commands such as ultrasound sensors acquisition and motor control. It is supported by the electronic device STM32F4D, which is a compact development system with lots of on-board peripherals, where the central part of the system is the 32-bit STM32F407VGT6 6ARM Cortes-M4-microcontroler. The microcontroller performs three basic tasks: 1) to communicate with the higher-level controller through USB; 2) reading encoder counts interrupt driven; and 3) generation of PWM duty cycle.
C. Control Strategy
The control task is to track a given time reference position and orientation with the minimum error. The block diagram of the mobile robot control system is shown in Fig. 1 . The control design consists of a cascade velocity and position feedback loop. The control position computes the velocity reference of each wheel. The decided velocity is controlled by the PID feedback control shown in Fig. 4 .
C. Control Strategy
As mentioned in the previous subsection, to measure the motor speed we used the incremental encoders coupled to the wheel axes. In order to filter the noise inherent to the measurement process (encoders) and performing its differentiation for obtaining a more reliable estimation of the wheels speed, a digital smoothing polynomial filter has been selected. The filter is based on a quadratic polynomial proposed by Savitzky-Golay [6] where each 4 raw data of each encoder at every ′ = 0.01 seconds ( ′ = 100 Hz) of sampling time are replaced by only one smoothed data using a second order polynomial. From this it is easy to obtain the derivative of the polynomial as an estimation of each wheel speed. In this case, the new sampling time is = 4 ′ ( = 25Hz) which is enough for the control loop. The formula of the Savitzky-Golay smoothing filter to estimate the speed of each wheel, i, is:
C. Control Strategy
where is the wheel speed, is the measure of the encoder with a sampling time of 
III. INTERVAL MODEL IDENTIFICATION
In many literature works [7] , mobile robots are modeled taking into account the inertia, traction, viscous friction and Coulomb friction forces, giving a non-linear and multivariable system. But in practice, the parameters of the resulting nonlinear model are difficult to quantify accurately, or the model itself is in error. Many techniques have been used to solve the problem of estimating the parameters or states of nonlinear systems. The most basic approach is to linearize the system around an operating point and to develop a first order set of linear state equations. Other techniques use least squares methods directly to try to minimize the errors in the non-linear estimation problem.
III. INTERVAL MODEL IDENTIFICATION
The technique used in this paper extends the linearization technique by using piece-wise linear models that cover the expected range of the state variables [8] . The only restriction is that the non-linearities must be able to be approximated as piecewise linear functions.
III. INTERVAL MODEL IDENTIFICATION
The non-linear model of the mobile robot platform can be written in the form:
III. INTERVAL MODEL IDENTIFICATION
where f and h are non-linear functions, ( ) ∈ ℛ 4 are the system state, ( ) ∈ ℛ 4 are the motor's voltage and ( ) ∈ ℛ 4 are the wheels velocities. The system is modeled as a set of piecewise linear equation of the form:
III. INTERVAL MODEL IDENTIFICATION
assuming that the non-linear model (2) is decomposed into q individual linear state space models (3) with matrices , and . Each linear model (3) is defined as a polyhedric partition ⊆ ℛ 4 of the state space. And, the equivalent inputoutput representation is given by:
III. INTERVAL MODEL IDENTIFICATION
where
III. INTERVAL MODEL IDENTIFICATION
In our study case, the set of linear systems has been identified performing system identification experiments around different equilibrium position under closed-loop control. Fig. 5 shows the voltage and velocity of one wheel when the closed-loop reference signal is a sequence of steps. Each step has been analyzed using the well-known AutoRegressive eXogenous (ARX) method, allowing the estimation of a set of individual systems. A first order model, with two parameters gain (k) and time constant (τ) (5), provides a good fit for the 20th individual linear models for each wheel.
III. INTERVAL MODEL IDENTIFICATION
The estimated parameters of (5) are given in Fig. 6 . Each color identifies a wheel of the robot. Notice that, every wheel has a different behavior, which is uncorrelated with the reference signal. Fig. 7 shows the simulated output of one of the estimated models together with the measured output. IV. ROBUST DESIGN USING QFT
III. INTERVAL MODEL IDENTIFICATION
As it has been mentioned in section III, the robot platform is characterized by multiple linear systems which makes it difficult to maintain good stability margins and performance properties for the closed-loop system. The purpose of this section is to introduce robust feedback control design in order to achieve improved robust performance of the controller.
A. QFT description
Quantitative feedback theory, developed by Isaac Horowitz [9] , (Fig. 8) , is a frequency domain design technique using the Nichols chart (NC) in order to achieve a desired robust design over a specified region of plant uncertainty. Specifications have to be expressed in frequency domain, thus desired time-domain responses have also to be translated into frequency domain tolerances which implies some constraints (bounds) on the loop transfer function in the NC.
A. QFT description
In general, an uncertain plant is described by the interval model:
A. QFT description
where N and D are polynomials in the s domain and θ is the vector of uncertain parameters of dimension n θ with their values bounded by a compact set θ ∈ Θ of box type, i.e., Θ = � ∈ ℛ � ≤ ≤ �. The feedback controller is defined as following
A. QFT description
where Nc and Dc are polynomials in the s domain; and is a vector controller parameters of dimension . Design process allows a designer to see what trade-offs are necessary to achieve a desired performance level. QFT technique procedure implies the next 4 basic steps: plant templates, frequency bounds, loop shaping and prefilter design (see [10] ).
Plant templates:
The uncertain plant P(s) is first translated in the frequency domain (using a discrete grid of frequencies Frequency Bounds: The description of the desired closedloop system performances (as robust stability, tracking, disturbance rejection, control effort) are required. In QFT design methodology, these requirements are described as frequency constraints on the nominal open-loop function and represented on the NC as curves, called frequency bounds. Prefilter Design: Finally, in the case of tracking performances, a prefilter, F(s), design is conducted using a Bode diagram. There exist different Computer Aided Design (CAD) packages to make the controller tuning easier. Some examples of these CAD packages are QFT Toolbox for Matlab [11] , QFT Control Toolbox (QFTCT) [12] and QFT Interactive Tool [13] .
B. Templates generation
There exist several approaches to solve the problem of generating conservative template boundaries [11] . In general these procedures try to envelope the plant uncertainty using some geometrical surface. In this paper, we propose to compute the templates taking into account the set of linear models (5) estimated in section III.
B. Templates generation
The plant templates are plotted on the Nichols chart at the useful desired frequency = {0.01, 0.1, 0.2, 1, 2, 10, 20, 50, 75} rad/seg for the estimated piecewise models (5).
B. Templates generation
The templates obtained for the family of plants and for the set of frequencies are as illustrated in Fig. 9 , where the nominal plant is for k =21.08 and τ = 1.053.
C. Performance specifications
Two performance specifications are formulated.
C. Performance specifications
Robust stability. To ensure robust stability in the closed loop system the next constraint must be satisfied:
C. Performance specifications
where ( ) is the bounded magnitude of the closed-loop frequency response and ( ) = ( ) ( ), is the open Fig. 8 . QFT generic control structure. Fig. 9 . Plant templates for the piecewise model. Control effort. The control effort specification is given by:
C. Performance specifications
where U (ω) is the bounded magnitude.
C. Performance specifications
For controller design, two numerical specifications have been considered to obtain a good compromise between robust stability and control effort by trial and error: = 1.2 (1.6 dB) and = 0.42. (-7.5 dB).
C. Performance specifications
The intersection of the QFT bounds ( Fig. 9 ) with the inequalities (8) and (9) was computed on the NC at the same frequencies of plant templates for the first specification, as depicted in Fig. 10 . The vertical curve line represents the frequency response of the nominal open-loop transfer function with the controller ( ) = 1 at several frequency points marked with the same colors as the bounds. The closed round lines represent the worst case of robust stability bounds and the lines in the lower area represent the worst case of control effort. The bound plotted with a solid line implies that 0 ( ) must lie above it in order to meet the performance specifications whereas the bound plotted with a dashed line implies that 0 ( ) must lie below [14] .
D. Loop-shaping controller
The objective is to design a controller G(s) with a PI structure modelled by
D. Loop-shaping controller
being the proportional gain and the integration gain. The controller is synthesized in NC by changing the gain and adding poles and zeros until the nominal plant lies near its bounds.
D. Loop-shaping controller
The transfer function of the controller designed considering both specifications is: 
V. PERFORMANCE VALIDATION
The validation of the obtained results are made checking the specifications in the frequency and temporal domains. The analysis of the closed loop system performances in frequency domain is illustrated in Fig. 12 , showing that the worst case closed loop frequency response and control effort magnitudes in both scenarios is below the specified values ( =1.2 = 1.584 dB and = 0.42=-7.5 dB, dashed line). Fig. 13 shows angular speed response and speed error of one wheel after introducing a set of step changes as speed reference in the four DC motors. In order to achieve a more detailed view of the angular speed response, Fig. 14 shows the four wheels angular speed evolution during two steps. Notice that the motors reach the reference speed rapidly with an overshoot. In all the scenario, the maximum overshoot obtained is less than 20% satisfying the design specification ( =1.2). The rise time is between 0.16 and 1.12s.
V. PERFORMANCE VALIDATION
The performance of the designed controller, G1, is compared with the original controller PID proposed by the platform developers, with parameters:
V. PERFORMANCE VALIDATION
=0.6 and =0.1, named as G2. The Figs. 15 and 16 show the results with the original controller, G2, applying the same set of step changes as speed reference in the four DC motors. In these results, there is no overshoot but the rise time exceeds 10 seconds. Finally, Table 1 displays a summary of the comparison regarding two well known criteria: integral absolute error (IAE) and integral absolute of controlled action (IAU) for the 4 wheels. The IAU is very similar in both controllers but the IAE is clearly reduced around 20% using the QFT controller with respect to the original one. In this paper the design and implementation of a robust speed controller for a four-wheeled omnidirectional mobile robot using QFT technique has been presented. The robot platform has been modeled with a piecewise model that has been estimated by means of least-squares identification using data collected from the real robot. The real experiments show the expected performances of the controller design and better dynamical performances with respect to the original controller of this mobile robot. 

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