Data‐driven iterative learning trajectory tracking control for wheeled mobile robot under constraint of velocity saturation

Abstract

Considering the wheeled mobile robot (WMR) tracking problem with velocity saturation, we developed a data‐driven iterative learning double loop control method with constraints. First, the authors designed an outer loop controller to provide virtual velocity for the inner loop according to the position and pose tracking error of the WMR kinematic model. Second, the authors employed dynamic linearisation to transform the dynamic model into an online data‐driven model along the iterative domain. Based on the measured input and output data of the dynamic model, the authors identified the parameters of the inner loop controller. The authors considered the velocity saturation constraints; we adjusted the output velocity of the WMR online, providing effective solutions to the problem of velocity saltation and the saturation constraint in the tracking process. Notably, the inner loop controller only uses the output data and input of the dynamic model, which not only enables the reliable control of WMR trajectory tracking, but also avoids the influence of inaccurate model identification processes on the tracking performance. The authors analysed the algorithm's convergence in theory, and the results show that the tracking errors of position, angle and velocity can converge to zero in the iterative domain. Finally, the authors used a simulation to demonstrate the effectiveness of the algorithm.

INTRODUCTION

Recently, with the rapid development of technology and science, wheeled mobile robots (WMRs) have been widely applied in military reconnaissance, industrial logistics, space exploration projects, and agricultural irrigation [1]. Trajectory tracking control is a basic problem to be solved in practical robot applications. Trajectory tracking can usually be split into two forms. One is path tracking, which only requires the robot to run according to the desired path without considering velocity. The other is real-time trajectory tracking, which requires the WMR to simultaneously track the position and velocity of the desired trajectory. The former cannot obtain the driving force, which makes its application impractical in practice, so real-time trajectory tracking control is more reasonable in actual requirements.

Researchers worldwide have used different methods to design various trajectory tracking controllers, such as sliding mode control [2–5], backstepping control [6], adaptive control [7–9], fuzzy control [10, 11], and intelligent control [12–17]. In Ref. [3], an antisaturation model-free adaptive integral terminal sliding mode control scheme is proposed, which combines an integral terminal sliding mode controller with model-free adaptive control and introduces a dynamic antisaturation compensator to solve the input saturation problem in the WMR trajectory tracking process. In Ref. [8], for the WMR trajectory tracking problem, an adaptive antisaturation tracking control scheme is proposed. A corresponding adaptive control law was designed considering system model parameter uncertainty and bounded disturbance, and the effect of input saturation is compensated by the auxiliary system to ensure the tracking effect of the system. The above control strategies are all asymptotically stable along the time domain. Although stable WMR trajectory tracking can be achieved, the control effect is often difficult to guarantee due to certain convergence processes in the initial stage of trajectory tracking. In Ref. [11], a hierarchical constraint method was used for the control of the mobile robot, with structural and kinematic constraints, and good control performance was obtained. In order to cope with uncertainty or unpredictable phenomena in robot modelling, a bimodal hybrid system based on stochastic differential equations was proposed in Ref. [12]. Velocity Following Control of a Pseudo-Driven Wheel for Reducing Internal Forces Between Wheels. In Ref. [13], the drive wheel is converted to a pseudo-drive wheel, and the robot body is controlled using kinematic constraints to design a speed-following control method that enables the robot to track a straight line well over loose terrain.

The iterative learning algorithm has been used to design WMR trajectory tracking control algorithms [18–22]. Iterative learning control is a valid control method for repeatedly completing time-bound control tasks [23]. However, the above work only studied the kinematic model of WMR, and the system dynamics model was not considered, including torque, driving force and mass, which may cause the degradation of tracking performance in actual engineering.

To solve the above problems, the data-driven control of tasks with dynamic models was introduced [24, 25]. Data-driven control is effective in the control of complex models. Data-driven control technology is already in use in many industries [26–30]. A model-free adaptive iterative learning fault-tolerant control for metro train tracking is presented in Ref. [27], where the concept of partial derivatives is first applied to transform the train dynamics model into a dynamic linearised model in compact form. The fault function is approximated from the radial basis function neural network under training and the output data of the faulty train system as compensation for the proposed scheme. Finally, the effectiveness of the proposed algorithm is verified by simulation results. However, it is uncommon in studies on WMR trajectory tracking. Additionally, the velocity saturation constraint must be considered in practical problems. Therefore, ensuring the desired effect of the system is often difficult if only the tracking performance index is considered and the problem of speed saturation is ignored when designing the control scheme.

For the dynamics and kinematics model of WMRs, in this study, we combined the benefits of both data-driven and iterative learning control in the presence of a velocity saturation constraint. We designed a double-loop control trajectory tracking control programme with constraints to achieve the real-time trajectory tracking with WMRs. The velocity was always within the prescribed constraint boundary. In addition, we provide theoretical proof of the practical application of high-performance WMR tracking. Compared with the existing control methods, this study's main contributions are as follows:

• The algorithm proposed in this paper is a fully data-driven control that does not rely on the robot's dynamics model, and the control signal is updated using only the system's input and output data. Compared with model-based control algorithms such as sliding mode control and model predictive control for mobile robots, it significantly reduces the reliance on the mechanistic model and improves the convenience in practical engineering.

• The tracking task is extended to iterative axes based on iterative learning to improve the current control performance by learning from the previous, which makes it possible to achieve full tracking in the time domain. Compared with the time-domain based robot control algorithm, it can effectively reduce the overshoot, oscillation and response time in the initial system tracking phase.

• The proposed algorithm is more feasible than the existing WMR iterative learning tracking control because the dynamics and kinematic models are considered simultaneously. Using the driving force as the control input can further lead to reasonable linear and angular velocities, and velocity saturation constraints are introduced to make the control constraints more reasonable.

The rest of this paper is organised as follows: problem formulation is given in Section 2. In Section 3, the control algorithm that is based on dynamic and kinematic models are designed, respectively, and then the convergence condition is analysed rigorously. A simulation example of WMR is given in Section 4, and some conclusions are given in Section 5.

PROBLEM FORMULATION

Consider a WMR with the mechanical structure shown in Figure 1. Define q  = [x,y,θ] ^T cartesian system, as the generalised position and the heading angle for the robot. Assume that there is no slip between the ground and the wheels as the robot moves. Therefore, the WMR has the following non-holonomic constraints: 1 $$\dot{x}\,\mathrm{sin}\,\theta -\dot{y}\,\mathrm{cos}\,\theta =\boldsymbol{A}{(\boldsymbol{q})}^{T}{\boldsymbol{\dot{q}}}=0,$$ where A (q) = [sin θ, −cos θ, 0] ^T denotes the constraint force vector. Under nonholonomic constraints, the kinematics model of a mobile robot can be expressed as folows: 2 $${\boldsymbol{\dot{q}}}=\boldsymbol{S}(\boldsymbol{q})\boldsymbol{u},$$ where u  = [ v , w ] denotes the robot's velocity, and S ( q ) denotes the transformation matrix function, which has the following form: 3 $$\boldsymbol{S}(\boldsymbol{q})=\left[\begin{array}{cc}\hfill \mathrm{cos}\,\theta \hfill & \hfill 0\hfill \\ \hfill \mathrm{sin}\,\theta \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right].$$

[IMAGE OMITTED. SEE PDF]

During the trajectory tracking process, constraints must be placed on the WMR's actual velocity, which must satisfy the following requirements for saturation: 4 $${\boldsymbol{u}}_{L}\le \boldsymbol{u}(k,i)\le {\boldsymbol{u}}_{U},$$ where u _L and u _U are minimum and maximum velocities, respectively.

If the controller for tracking the trajectory is designed solely based on the model that came before it, the problem of tracking the trajectory of a WMR is only solved at the kinematics level. To further improve tracking accuracy, the influence of mechanical characteristics, such as system mass and moment of inertia, on the control performance must be considered. Hence, a dynamic model in the Euler–Lagrange form is required, which is as follows: 5 $$\boldsymbol{M}{\boldsymbol{\ddot{q}}}+\boldsymbol{C}({\boldsymbol{\dot{q}}}){\boldsymbol{\dot{q}}}+\boldsymbol{G}=\boldsymbol{E}(\boldsymbol{q})\boldsymbol{\tau }+\boldsymbol{A}(\boldsymbol{q})\gamma ,$$ where $$\boldsymbol{\tau }={\left[{\boldsymbol{\tau }}_{1},{\boldsymbol{\tau }}_{2}\right]}^{T}\in {R}^{2}$$ represents the control torque that is imposed on the wheel; M  ∈ R ^{3 × 3} is an inertia matrix; $$\boldsymbol{C}({\boldsymbol{\dot{q}}})\in {R}^{3\times 3}$$ is the centrifugal force and Coriolis force matrix; G  ∈ R ³ represents the gravitational vector, given that the WMR is moving horizontally, G  = 0; E ( q ) ∈ R ^{3 × 2} denotes the input matrix; γ is the constraint force vector. The matrix parameters shown above have the same precise shape as those in Ref. [26].

To eradicate Lagrange multipliers, substitute Equation (2) into Equation (5), and multiply the left-hand side of model (5) by S ^T ( q ); the relationship between the control torque and the WMR velocity is obtained as follows: 6 $${\boldsymbol{S}}^{T}\boldsymbol{M}\boldsymbol{S}{\boldsymbol{\dot{u}}}+{\boldsymbol{S}}^{T}\boldsymbol{M}\dot{\boldsymbol{S}}\boldsymbol{u}+{\boldsymbol{S}}^{T}\boldsymbol{C}\dot{\boldsymbol{S}}\boldsymbol{u}={\boldsymbol{S}}^{T}\boldsymbol{E}\boldsymbol{\tau }+{\boldsymbol{S}}^{T}\boldsymbol{A}\gamma $$ where S ^T ( q ) A ( q ) = 0. For the sake of simplicity, we redefine $${\overline{\boldsymbol{M}}}_{1}={\boldsymbol{S}}^{T}\boldsymbol{M}\boldsymbol{S},{\overline{\boldsymbol{M}}}_{2}={\boldsymbol{S}}^{T}\boldsymbol{M}\dot{\boldsymbol{S}}$$, $$\overline{\boldsymbol{C}}={\boldsymbol{S}}^{T}\boldsymbol{C}\boldsymbol{S},\overline{\boldsymbol{E}}={\boldsymbol{S}}^{T}\boldsymbol{E}$$. The further expression of the dynamic equation is as follows: 7 $${\overline{\boldsymbol{M}}}_{1}{\boldsymbol{\dot{u}}}+{\overline{\boldsymbol{M}}}_{2}\boldsymbol{u}+\overline{\boldsymbol{C}}\boldsymbol{u}=\overline{\boldsymbol{E}}\tau .$$

For sampling time ΔT, the discrete form of the dynamic and kinematic model at the moment k is 8 $$\boldsymbol{q}(k+1,i)=\boldsymbol{q}(k,i)+{\Delta }T\boldsymbol{S}(\boldsymbol{q}(k,i))\boldsymbol{u}(k,i),$$ 9 $$\begin{array}{l}\boldsymbol{u}(k+1,i)=\left(\boldsymbol{I}-\frac{{\Delta }T\left({\overline{\boldsymbol{M}}}_{2}(k,i)+\boldsymbol{C}(k,i)\right)}{{\overline{\boldsymbol{M}}}_{1}(k,i)}\right)\boldsymbol{u}(k,i)\hfill \\ +\frac{{\Delta }T\overline{\boldsymbol{E}}(k,i)}{{\overline{\boldsymbol{M}}}_{1}(k,i)}\boldsymbol{u}(k,i)\hfill \end{array}$$ where i is the number of times the process is repeated, and k ∈ [0, T] is the time interval.

Our aim in this study was to design a method of control that is based on the dynamic and kinematic models of a WMR. In other words, the robot needs to track not only the trajectory, but also the path's angular and linear velocities, that is, 10 $$\underset{k\to \infty }{\mathrm{lim}\,}\left\Vert \boldsymbol{q}(k,i)-{\boldsymbol{q}}_{d}(k)\right\Vert =0,\underset{k\to \infty }{\mathrm{lim}\,}\left\Vert {\boldsymbol{u}}_{d}(k)-\boldsymbol{u}(k,i)\right\Vert =0.$$

CONTROLLER DESIGN

A detailed description of the control problem was given in the previous section. Next, according to the kinematic model (8) and the dynamic model (9), a control strategy for tracking the WMR trajectory is designed.

• The outer loop control scheme is designed according to the kinematics model (8) of the system, and the virtual reference velocity of the inner loop control scheme is obtained;

• An inner loop control scheme is constructed according to the dynamic model (9) of the system. A data-driven control scheme is introduced, and the dynamic model is transformed into a data-driven model. According to the model's input and output data and the velocity saturation constraint condition, the output velocity signal is modified online to track the desired velocity at each time point. The design of the control scheme is shown in Figure 2.

[IMAGE OMITTED. SEE PDF]

Kinematic control

Firstly, the outer loop control scheme is conceived on the basis of the WMR kinematics model (8). The expected track direction of the definite system follows is $${\boldsymbol{q}}_{d}={\left[\begin{array}{ccc}\hfill {x}_{d}\hfill & \hfill {y}_{d}\hfill & \hfill {\theta }_{d}\hfill \end{array}\right]}^{T}\in {R}^{3}$$, then the expected trajectory model can be described as 11 $${\boldsymbol{q}}_{d}(k+1)={\boldsymbol{q}}_{d}(k)+{\Delta }T\boldsymbol{S}\left({\boldsymbol{q}}_{d}(k)\right){\boldsymbol{u}}_{d}(k)$$ where $${\boldsymbol{u}}_{d}={\left[\begin{array}{cc}\hfill {v}_{d}\hfill & \hfill {w}_{d}\hfill \end{array}\right]}^{T}\in {R}^{2}$$ denotes the desired velocity.

The WMR kinematics equations and the desired trajectory model (11) comply with the following properties and assumptions:

$1.$

Property

The matrix S ( q (k, i)) is bounded, satisfying $$\left\Vert \boldsymbol{S}\left({\boldsymbol{q}}_{d}(k)\right)\right\Vert \le {b}_{B}$$, where b _B is a positive constant, S ( q (k, i)) is a full rank matrix for q (k, i).

$2.$

Property

The matrix function S ( q (k, i)) satisfies the generalised Lipschitz condition, that is to say, there is a normal number l _b for $$\left\Vert \boldsymbol{S}\left({\boldsymbol{q}}_{1}(k,i)\right)-\boldsymbol{S}\left({\boldsymbol{q}}_{2}(k,i)\right)\right\Vert \le {l}_{b}\left\Vert {\boldsymbol{q}}_{1}(k,i)-{\boldsymbol{q}}_{2}(k,i)\right\Vert $$, ∀k ∈ [0, T].

$1.$

Assumption

The initial position error fulfils q (0, i) =  q _d (0).

$2.$

Assumption

The velocity of virtual tracking target is bounded here, $$\underset{1\le k\le T}{\mathrm{max}\,}\left\Vert {\boldsymbol{u}}_{d}(k)\right\Vert \le {b}_{ud}$$ is a positive constant.

$1.$

Remark

Assumption 1 is a basic requirement of iterative learning control, which is also a must in the iterative learning-based mobile robot control [15–18]. For robots performing repetitive motion tasks, the condition of identical initial positions is very easy to satisfy. Assumption 2 requires that the virtual target velocity of the robot needs to be bounded in order to satisfy its output bounded, which is reasonable in practice. From an energy point of view, the kinetic energy of a mechanical system is always positive, and finite inputs necessarily produce finite outputs.

Defining the tracking error as q _e (k, i) =  q _d (k) −  q (k, i). Assume that u _r (k, i) is the output velocity model of WMR kinematics. The outer loop control scheme using iterative learning algorithm is designed as follows: 12 $${\boldsymbol{u}}_{r}(k,i)={\boldsymbol{u}}_{r}(k,i)+{\boldsymbol{\Gamma }}_{1}(k){\boldsymbol{q}}_{e}(k+1,i)+{\boldsymbol{\Gamma }}_{2}(k){\boldsymbol{q}}_{e}(k,i+1),$$ where Γ₁(k) and Γ₂(k) are the matrices of learning gain parameters, which satisfies $$\left\Vert {{\Gamma }}_{1}(k)\right\Vert \le {b}_{{\Gamma }1},\left\Vert {{\Gamma }}_{2}(k)\right\Vert \le {b}_{{\Gamma }2},k\in [0,T],{b}_{{\Gamma }1},{b}_{{\Gamma }2}$$ are two positive constants.

$1.$

Theorem

Consider the discrete-time kinematics ( 8 ) of the system satisfying Assumptions 1 and 2 for all q (k, i) ∈ R ⁿ , if the parameter of control algorithm ( 12 ) satisfies 13 $$\left\Vert \boldsymbol{I}-{\boldsymbol{\Gamma }}_{1}(k)\boldsymbol{S}(\boldsymbol{q}(k,i))\right\Vert < 1,$$ then when k ∈ [0, T] and i → ∞, u _r (k, i), q (k, i) converge to u _d (k), q _d (k) respectively. And $$\left\Vert {\boldsymbol{u}}_{d}(k)-\boldsymbol{u}(k,i)\right\Vert ,\left\Vert {\boldsymbol{q}}_{d}(k)-\boldsymbol{q}(k,i)\right\Vert $$ are bounded, that is, $${\mathrm{lim}}_{i\to \infty }\left\Vert {\boldsymbol{u}}_{r}(k,i)-{\boldsymbol{u}}_{d}(k)\right\Vert =0$$.

The proof of Theorem 1 can be seen in Ref. [15].

$2.$

Remark

The large amount of data generated during the operation of the robot contains all the useful information about the operation and state of the robot. Through the dynamic linearisation technique, the input/output data relationship of the system is transformed into an equivalent iterative incremental data model. As an inherent feature of WMR itself, the input and output of the system are all data affected by the constraint, and the data relationship must contain the feature of non-completeness constraint. Therefore, the data-driven iterative learning WMR tracking control based on model-free adaptation used in this paper does not lose the non-holism constraint of the robot.

Dynamic control

The dynamic model (9) contains a large number of time-varying measurement parameters. To avoid the influence of these factors on the trajectory tracking effect of the WMR, in the design of inner loop controller, the dynamic linearisation technique is applied to transform the model into an online data model along the iterative domain. The inner control scheme is then designed by combining the iterative learning control and data-driven control.

The partial derivative of u (k, i + 1) with respect to output τ (k, i) is continuous, as is visible from the WMR discrete dynamics model (9). The generalised Lipschitz condition is satisfied along the iteration axis of this model, that is, if ‖Δ τ (k, i)‖ ≠ 0, then 14 $$\Vert {\Delta }\boldsymbol{u}(k+1,i)\Vert \le b\Vert {\Delta }\boldsymbol{\tau }(k,i)\Vert ,\forall k\in [0,T],i=1,2,\text{\ldots }$$ where Δ u (k, i) =  u (k, i) −  u (k, i − 1) and Δ τ (k, i) =  τ (k, i) −  τ (k, i − 1), b < ∞ is a positive Lipschitz constant.

$3.$

Remark

It is a common restriction for non-linear systems that the system's partial derivative with regard to the variable in question is continuous. This condition can guarantee that the rate of change of outputs of non-linear systems will not cause escape in finite time.

From Equation (9), the dynamic model relation along the iterative domain is obtained as follows: 15 $${\Delta }\boldsymbol{u}(k+1,i)=\boldsymbol{u}(k+1,i)-\boldsymbol{u}(k+1,i-1)=\left[\left(\boldsymbol{I}-\frac{{\Delta }T\left({\overline{\boldsymbol{M}}}_{2}(k,i)+\boldsymbol{C}(k,i)\right)}{{\overline{\boldsymbol{M}}}_{1}(k,i)}\right)\boldsymbol{u}(k,i)+\frac{{\Delta }T\overline{\boldsymbol{E}}(k,i)}{{\overline{\boldsymbol{M}}}_{1}(k,i)}\boldsymbol{\tau }(k,i)\right]-\left[\left(\boldsymbol{I}-\frac{{\Delta }T\left({\overline{\boldsymbol{M}}}_{2}(k,i)+\boldsymbol{C}(k,i)\right)}{{\overline{\boldsymbol{M}}}_{1}(k,i)}\right)\boldsymbol{u}(k,i)+\frac{{\Delta }T\overline{\boldsymbol{E}}(k,i)}{{\overline{\boldsymbol{M}}}_{1}(k,i)}\boldsymbol{\tau }(k,i-1)\right]+\left[\left(\boldsymbol{I}-\frac{{\Delta }T\left({\overline{\boldsymbol{M}}}_{2}(k,i)+\boldsymbol{C}(k,i)\right)}{{\overline{\boldsymbol{M}}}_{1}(k,i)}\right)\boldsymbol{u}(k,i)+\frac{{\Delta }T\overline{\boldsymbol{E}}(k,i)}{{\overline{\boldsymbol{M}}}_{1}(k,i)}\boldsymbol{\tau }(k,i-1)\right]-\left[\begin{array}{c}\hfill \left(\boldsymbol{I}-\frac{{\Delta }T\left({\overline{\boldsymbol{M}}}_{2}(k,i-1)+\boldsymbol{C}(k,i-1)\right)}{{\overline{\boldsymbol{M}}}_{1}(k,i-1)}\right)\boldsymbol{u}(k,i-1)\hfill \\ \hfill +\frac{{\Delta }T{\overline{\boldsymbol{E}}}_{1}(k,i-1)}{{\overline{\boldsymbol{M}}}_{1}(k,i-1)}\boldsymbol{\tau }(k,i-1)\hfill \end{array}\right]$$

Therefore, according to the theorem of differential mean value, Equation (15) is obtained as follows 16 $${\Delta }\boldsymbol{u}(k+1,i)=\frac{\partial \boldsymbol{u}(k+1,i)}{\partial \boldsymbol{\tau }(k,i)}{\Delta }\boldsymbol{\tau }(k,i)+\boldsymbol{\psi }(k,i),$$ where $$\frac{\partial \boldsymbol{u}(k+1,i)}{\partial \boldsymbol{\tau }(k,i)}$$ denotes the optimal partial derivative of u (k + 1, i) with respect to input τ (k, i) at some point between τ (k, i) and τ (k, i − 1).

For each moment k of each iteration i, consider the following equation containing h (k, i) as follows: 17 $$\boldsymbol{\psi }(k,i)=\boldsymbol{h}(k,i){\Delta }\boldsymbol{\tau }(k,i).$$

Due to ‖Δ τ (k, i)‖ ≠ 0, Equation (17) must have some unique solution h *(k, i), and let $$\boldsymbol{\phi }(k,i)=\frac{\partial \boldsymbol{u}(k+1,i)}{\partial \boldsymbol{\tau }(k,i)}+{\boldsymbol{h}}^{\ast }(k,i)$$ Then, one has 18 $${\Delta }\boldsymbol{u}(k+1,i)=\boldsymbol{\phi }(k,i){\Delta }\boldsymbol{\tau }(k,i),$$ where ϕ (k, i) ∈ R ^n × n is called Pseudo Partial Derivatives (PPD). According Equation (14), ϕ (k, i) is bounded.

The WMR dynamics model's equivalent linearised form is obtainable from Equation (18), that is, WMR dynamics' data-driven model: 19 $$\boldsymbol{u}(k+1,i)=\boldsymbol{u}(k+1,i-1)+\boldsymbol{\phi }(k,i){\Delta }\boldsymbol{\tau }(k,i)$$

$4.$

Remark

In dynamic linearisation, it is required to meet ‖Δ τ (k, i) ≠ 0‖. In practice, if the case ‖Δ τ (k, i) = 0‖ occurs at a certain iteration and sampling time, a new dynamic linearisation can be employed after shifting σ _i  ∈ Z⁺ iteration instants until τ (k, i) ≠  τ (k, i − σ) holds.

Defining u _r (k) as the virtual reference velocity after i learning iterations. Considering the data-driven model of dynamics, a data-driven iterative learning control algorithm with speed saturation constraints is designed below.

The velocity saturation constraint of WMR can be described as 20 $${\boldsymbol{u}}^{\prime }(k,i)=\mathrm{s}\mathrm{a}\mathrm{t}(\boldsymbol{u}(k,i))=\left\{\begin{array}{ll}{\boldsymbol{u}}_{U}\quad \hfill & \boldsymbol{u}(k,i)\ge {\boldsymbol{u}}_{U}\hfill \\ \boldsymbol{u}(k,i)\quad \hfill & {\boldsymbol{u}}_{U}\le \boldsymbol{u}(k,i)\le {\boldsymbol{u}}_{L}\hfill \\ {\boldsymbol{u}}_{L}\quad \hfill & \boldsymbol{u}(k,i)\le {\boldsymbol{u}}_{L}\hfill \end{array}\right.$$

Defining a new error variable e (k, i) = u _r (k) − u (k, i). Then, the actual velocity tracking error of WMR is 21 $${\boldsymbol{e}}^{\prime }(k,i)={\boldsymbol{u}}_{r}(k)-{\boldsymbol{u}}^{\prime }(k,i).$$

Consider the following criterion function with velocity saturation constraints: 22 $$J(\boldsymbol{\tau }(k,i))={\left\Vert {\boldsymbol{e}}^{\prime }(k+1,i)\right\Vert }^{2}+\lambda \Vert \boldsymbol{\tau }(k,i)-\boldsymbol{\tau }(k,i-1){\Vert }^{2}$$ where λ > 0 is a weight factor. Based on the optimal condition ∂J/∂ τ (k, i) = 0, the control input updating scheme of inner loop control algorithm can be described as: 23 $$\boldsymbol{\tau }(k,i)=\boldsymbol{\tau }(k,i-1)+\frac{\rho {\boldsymbol{\phi }}^{T}(k,i)}{\lambda +\Vert \boldsymbol{\phi }(k,i){\Vert }^{2}}{\boldsymbol{e}}^{\prime }(k+1,i-1)$$ where ρ ∈ (0, 1] is a step factor.

The parameter ϕ (k, i) in Equation (23) is unknown, hence requiring an update law to estimate ϕ (k, i)’s value online. Define the following parameter estimation criterion metric function: 24 $$\begin{array}{rl}\hfill J(\boldsymbol{\phi }(k,i))& ={\left\Vert {\Delta }{\boldsymbol{u}}^{\prime }(k+1,i-1)-\boldsymbol{\phi }(k,i){\Delta }\boldsymbol{\tau }(k,i-1)\right\Vert }^{2}\hfill \\ \hfill & \quad +\mu \Vert \boldsymbol{\phi }(k,i)-\widehat{\boldsymbol{\phi }}(k,i-1){\Vert }^{2},\hfill \end{array}$$ where μ > 0 is a step factor.

By differentiating J( ϕ (k, i)) with respect to ϕ (k, i) and making it zero, the following parameter update algorithm is obtained as follows: 25 $$\begin{array}{rl}\hfill & \widehat{\boldsymbol{\phi }}(k,i)=\widehat{\boldsymbol{\phi }}(k,i-1)+\frac{\eta }{\mu +\Vert {\Delta }\boldsymbol{\tau }(k,i-1){\Vert }^{2}}\hfill \\ \hfill & \quad \quad \quad \quad \times \left({\Delta }{\boldsymbol{u}}^{\prime }(k+1,i-1)-{\widehat{\boldsymbol{\phi }}}^{T}(k,i-1){\Delta }\boldsymbol{\tau }(k,i-1)\right)\hfill \\ \hfill & \quad \quad \quad \quad \times \hspace*{.5em}{\Delta }{\boldsymbol{\tau }}^{T}(k,i-1),\hfill \end{array}$$ where η ∈ (0, 1] is a step factor. $$\widehat{\boldsymbol{\phi }}(k,i)$$ is the estimation value of ϕ (k, i).

Based on the estimation algorithm (25), the control input update scheme (23) becomes 26 $$\boldsymbol{\tau }(k,i)=\boldsymbol{\tau }(k,i-1)+\frac{\rho {\widehat{\boldsymbol{\phi }}}^{T}(k,i)}{\lambda +\Vert \widehat{\boldsymbol{\phi }}(k,i){\Vert }^{2}}{\boldsymbol{e}}^{\prime }(k+1,i-1)$$

Here is the reset algorithm that makes sure the parameter estimation algorithm is stable, 27 $$\begin{array}{rl}\hfill & \widehat{\boldsymbol{\phi }}(k,i)=\widehat{\boldsymbol{\phi }}(k,1),\hspace*{.5em}\text{if}\hspace*{.5em}\vert \widehat{\boldsymbol{\phi }}(k,i)\vert \le \varepsilon \hfill \\ \hfill & \hspace*{.5em}\quad \quad \quad \hspace*{.5em}\text{or}\hspace*{.5em}\mathrm{sign}\left\{\widehat{\boldsymbol{\phi }}(k,i)\right\}\ne \mathrm{sign}\left\{\widehat{\boldsymbol{\phi }}(k,1)\right\},\hfill \end{array}$$ where ɛ > 0 is a small positive constant, $$\widehat{\boldsymbol{\phi }}(k,1)$$ is the initial value of $$\widehat{\boldsymbol{\phi }}(k,i)$$.

$5.$

Remark

It is important to notice that the reset algorithm (27) helps to improve the PPD estimation algorithm's tracking capability for time-varying parameters. It is possible to observe, all that is needed for the control input procedure and the parameter estimation arithmetic is the I/O data from the system dynamics model at a given time. It does not use any parameters of the dynamic model itself, which can reduce the impact of inaccurate tracking performance effectively.

$6.$

Remark

This section proposes an outer loop control scheme containing a control input update algorithm (26), reset algorithm (27) and parameter estimation algorithm (25) based on a dynamic system that can be turned into a data-driven model (19).

For the sake of the rigour of the discussion, the following assumption is given:

$3.$

Assumption

For any k ∈ [0, T] and i = 1, 2, …, the signal of element of ϕ (k, i) is consistent, that is, $$\boldsymbol{\phi }(k,i) > \bar{\varepsilon } > 0$$ or $$\boldsymbol{\phi }(k,i)< -\bar{\varepsilon }< 0$$, where $$\bar{\varepsilon }$$ is a small positive constant. This paper considers that $$\boldsymbol{\phi }(k,i) > \bar{\varepsilon } > 0$$ without loss of generality.

$7.$

Remark

The physical significance of Assumption 3 is obvious: when the control input increases, the corresponding controlled system output should be non-decreasing, which can be considered as a ‘quasi-linear’ feature of the system. This condition is similar to the assumption that the control direction is known or constant sign in the model-based control method, and the robot control system satisfies this assumption.

$2.$

Theorem

For the system (19), if the partial derivative of u (k, i) is continuous with respect to the control torque τ (k, i) and the system is generalised Lipschitz, the tracking error of the system exists and converges to zero, that is, lim _i→∞ ‖ e (k + 1, i)‖ = 0.

$8.$

Remark

The initial output τ (i, 0) at the initial moment of each iteration of the system is random, but it is limited.

Firstly, the boundedness of PPD estimation value $$\widehat{\boldsymbol{\phi }}(k,i)$$.

Defining $$\tilde{\boldsymbol{\phi }}(k,i)=\widehat{\boldsymbol{\phi }}(k,i)-\boldsymbol{\phi }(k,i)$$ as the estimation error of $$\widehat{\boldsymbol{\phi }}(k,i)$$. Subtracting ϕ (k, i) from both sides of Equation (25), the following equation can be obtained: 28 $$\begin{array}{l}\tilde{\boldsymbol{\phi }}(k,i)=\tilde{\boldsymbol{\phi }}(k,i-1)-(\boldsymbol{\phi }(k,i)-\boldsymbol{\phi }(k,i-1))\hfill \\ +\frac{\eta \left({\Delta }\boldsymbol{u}(k,i-1)-{\widehat{\boldsymbol{\phi }}}^{T}(k,i-1){\Delta }\boldsymbol{\tau }(k,i-1)\right)}{\mu +\Vert {\Delta }\boldsymbol{\tau }(k,i-1){\Vert }^{2}}\hfill \\ \times {\Delta }{\boldsymbol{\tau }}^{T}(k,i-1)\hfill \\ =\tilde{\boldsymbol{\phi }}(k,i-1)-{\Delta }\boldsymbol{\phi }(k,i)-\frac{\eta \tilde{\boldsymbol{\phi }}(k,i-1){\Delta }\boldsymbol{\tau }(k,i-1)}{\mu +\Vert {\Delta }\boldsymbol{\tau }(k,i-1){\Vert }^{2}}\hfill \\ \times {\Delta }{\boldsymbol{\tau }}^{T}(k,i-1),\hfill \end{array}$$ where Δ ϕ (k, i) =  ϕ (k, i) −  ϕ (k, i − 1). Taking the mathematic expectation of Equation (28), then it follows 29 $$\begin{array}{rl}\hfill & \Vert \tilde{\phi }(k,i)\Vert =\Vert \tilde{\phi }(k,i-1)\Vert \left(I-\frac{\eta \Vert {\Delta }\tau (k,i-1){\Vert }^{2}}{\mu +\Vert {\Delta }\tau (k,i-1){\Vert }^{2}}\right)\hfill \\ \hfill & \quad \quad \quad \quad \enspace+\Vert {\Delta }\phi (k,i)\Vert .\hfill \end{array}$$

The previous analysis shows that ‖ ϕ (k, i)‖ has an upper bound. Supposing $$\Vert \boldsymbol{\phi }(k,i)\Vert \le \bar{b}$$, then $$\Vert \boldsymbol{\phi }(k,i)-\boldsymbol{\phi }(k,i-1)\Vert \le 2\bar{b}$$ holds. Due to 0 < η ≤ 1, μ > 0, it is obvious that $$\left\Vert \boldsymbol{I}-\frac{\eta \Vert {\Delta }\boldsymbol{\tau }(k,i-1){\Vert }^{2}}{\mu +\Vert {\Delta }\boldsymbol{\tau }(k,i-1){\Vert }^{2}}\right\Vert \in (0,1)$$. There exists a constant d ₁ satisfies 30 $$0\le \left\Vert \boldsymbol{I}-\frac{\eta \Vert {\Delta }\boldsymbol{\tau }(k,i-1){\Vert }^{2}}{\mu +\Vert {\Delta }\boldsymbol{\tau }(k,i-1){\Vert }^{2}}\right\Vert \le 1-\frac{\eta {\varepsilon }^{2}}{\mu +{\varepsilon }^{2}}={d}_{1}< 1$$

It is important to emphasise that this argument only needs to prove the existence of d ₁, not its exact value.

Simply replacing Equation (30) with Equation (29) can obtain 31 $$\Vert \tilde{\boldsymbol{\phi }}(k,i)\Vert \le {d}_{1}\Vert \tilde{\boldsymbol{\phi }}(k,i-1)\Vert +2\bar{b}\le {d}_{1}^{i-1}\Vert \tilde{\boldsymbol{\phi }}(k,1)\Vert +\frac{2\bar{b}}{1-{d}_{1}}$$

According to Equation (31), it is obvious that $$\tilde{\boldsymbol{\phi }}(k,i)$$ and $$\widehat{\boldsymbol{\phi }}(k,i)$$ are bounded.

Afterwards, we will examine the velocity tracking error e (k + 1, i) convergence analysis.

Denote 32 $$\boldsymbol{r}(k,i)=\left\{\begin{array}{ll}\frac{{\boldsymbol{u}}_{r}(k)-{\boldsymbol{u}}_{U}}{\boldsymbol{e}(k,i)}\enspace\hfill & \boldsymbol{e}(k,i) > {\boldsymbol{u}}_{r}(k)-{\boldsymbol{u}}_{U}\hfill \\ 1\enspace\hfill & {\boldsymbol{u}}_{r}(k)-{\boldsymbol{u}}_{U}\le \boldsymbol{e}(k,i)\le {\boldsymbol{u}}_{r}(k)-{\boldsymbol{u}}_{L},\hfill \\ \frac{{\boldsymbol{u}}_{r}(k)-{\boldsymbol{u}}_{L}}{\boldsymbol{e}(k,i)}\enspace\hfill & \boldsymbol{e}(k,i) > {\boldsymbol{u}}_{r}(k)-{\boldsymbol{u}}_{L}\hfill \end{array}\right.$$ then one has 33 $${\boldsymbol{e}}^{\prime }(k,i)=\boldsymbol{r}(k,i)\boldsymbol{e}(k,i).$$

Substituting Equation (23) into Equation (33) and taking absolute values, 34 $$\left\Vert {\boldsymbol{e}}^{\prime }(k+1,i)\right\Vert =\Vert \boldsymbol{r}(k+1,i)\Vert \Vert \boldsymbol{e}(k+1,i)\Vert =\Vert \boldsymbol{r}(k+1,i)\Vert \left\Vert {\boldsymbol{u}}_{r}(k+1)-\boldsymbol{u}(k+1,i)\right\Vert =\Vert \boldsymbol{r}(k+1,i)\Vert \left\Vert {\boldsymbol{u}}_{r}(k+1)-\boldsymbol{u}(k+1,i-1)-\boldsymbol{\phi }(k,i){\Delta }\boldsymbol{\tau }(k,i)\right\Vert =\Vert \boldsymbol{r}(k+1,i)\Vert \Vert \boldsymbol{e}(k+1,i-1)-\boldsymbol{\phi }(k,i){\Delta }\boldsymbol{\tau }(k,i)\Vert \le \Vert \boldsymbol{r}(k+1,i)\Vert \left\Vert \boldsymbol{I}-\frac{\rho \boldsymbol{\phi }(k,i){\widehat{\boldsymbol{\phi }}}^{T}(k,i)}{\boldsymbol{\lambda }+\Vert \widehat{\boldsymbol{\phi }}(k,i)\Vert }\right\Vert \Vert \boldsymbol{e}(k+1,i-1)\Vert $$

If $${\lambda }_{\mathrm{min}}=\frac{\bar{b}}{4}$$ holds, since $$\boldsymbol{\phi }(k,i){\widehat{\boldsymbol{\phi }}}^{T}(k,i) > 0$$ and ϕ (k, i) is bounded, there exists a positive constant M _d  < 1 such that when λ > λ _min, the following inequality is satisfied: 35 $$0< {M}_{d}< \frac{\boldsymbol{\phi }(k,i){\widehat{\boldsymbol{\phi }}}^{T}(k,i)}{\lambda +\Vert \widehat{\boldsymbol{\phi }}(k,i){\Vert }^{2}}\le \frac{\bar{b}{\widehat{\boldsymbol{\phi }}}^{T}(k,i)}{2\sqrt{\lambda }\vert \widehat{\boldsymbol{\phi }}(k,i)\vert }< \frac{\bar{b}}{2\sqrt{{\lambda }_{\mathrm{min}}}}=1.$$

From Equation (35), when 0 < ρ < 1 and λ > λ _min, there must be a constant that satisfies d ₂ < 1 so that 36 $$\begin{array}{rl}\hfill & \left\Vert \boldsymbol{I}-\frac{\rho \boldsymbol{\phi }(k,i){\widehat{\phi }}^{T}(k,i)\Vert }{\lambda +\Vert \widehat{\boldsymbol{\phi }}(k,i){\Vert }^{2}}\right\Vert =1-\frac{\rho \boldsymbol{\phi }(k,i){\widehat{\boldsymbol{\phi }}}^{T}(k,i)}{\lambda +\Vert \widehat{\boldsymbol{\phi }}(k,i){\Vert }^{2}}\hfill \\ \hfill & \quad \quad \quad \quad \quad \quad \quad \quad \quad \le 1-\rho {M}_{d}\triangleq {d}_{2}< 1.\hfill \end{array}$$

Combining Equations (34) and (36), we can get 37 $$\Vert \boldsymbol{e}(k+1,i)\Vert \le {d}_{2}\Vert \boldsymbol{e}(k+1,i-1)\Vert ,$$ then 38 $$\Vert \boldsymbol{e}(k+1,i)\Vert \le {d}_{2}^{i-1}\Vert \boldsymbol{e}(k+1,1)\Vert .$$

Hence, we can have e (k + 1, i) converges monotonically when the number of iterations i tends to infinity, and from this it can see 39 $$\underset{i\to \infty }{\mathrm{lim}\,}\left\Vert {\boldsymbol{e}}^{\prime }(k+1,i)\right\Vert =\underset{i\to \infty }{\mathrm{lim}\,}\frac{\Vert \boldsymbol{e}(k+1,i)\Vert }{\Vert \boldsymbol{r}(k+1,i)\Vert }=0.$$

Combining Theorem 1, $${\mathrm{lim}}_{i\to \infty }\left\Vert {\boldsymbol{u}}_{d}(k)-\boldsymbol{u}(k,i)\right\Vert =0$$ can be obtained.

This is the end of the proof.

SIMULATION EXAMPLE

In this section, the proposed scheme's effectiveness is demonstrated by a WMR trajectory that tracks simulation. It is worth pointing out that the control approach does not make use of any of the model information of WMR dynamics. The simulation model is given to generate the system's I/O data only and does not participate in control algorithm's design. The selection of WMR model's physical parameters is the same as that of Ref. [26]: $$\begin{array}{rl}\hfill & m=15\hspace*{.5em}\mathrm{k}\mathrm{g},I=10{\mathrm{k}\mathrm{g}\mathrm{m}}^{2},R=0.15\hspace*{.5em}\mathrm{m},r=0.1\hspace*{.5em}\mathrm{m},\hfill \\ \hfill & \boldsymbol{M}=\mathrm{diag}([\mathrm{m},\mathrm{m},I]),{\overline{\boldsymbol{M}}}_{1}=\mathrm{diag}([m,I]),\hfill \\ \hfill & {\overline{\boldsymbol{M}}}_{2}={\boldsymbol{S}}^{T}\boldsymbol{M}\dot{\boldsymbol{S}}=0,\overline{\boldsymbol{E}}=\frac{1}{r}\left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill -R\hfill & \hfill R\hfill \end{array}\right],\hfill \end{array}$$

For calculation's ease assume the Coriolis force C = 0.

The reference trajectory of WMR is 40 $$\left\{\begin{array}{ll}{x}_{d}=\mathrm{cos}((k-1)\times \pi \times {\Delta }T)\quad \hfill & k=1,2,\text{\ldots },100\hfill \\ {y}_{d}=\mathrm{sin}((k-1)\times \pi \times {\Delta }T)\quad \hfill & k=1,2,\text{\ldots },100\hfill \\ {\theta }_{d}={\Delta }T\times \pi \times (k-1)+\pi /2\quad \hfill & k=1,2,\text{\ldots },100\hfill \end{array}\right.$$ where the sampling time is ΔT = 0.001 s. The WMR's initial pose is [10π/2] ^T , the initial values of the angular velocity and the linear velocity are [6, 3] ^T , the initial values of the PPD estimate are $$\widehat{\boldsymbol{\phi }}=\mathrm{diag}([0.5,0.5])$$. Proposed control algorithm parameters are selected as follows after repeated experiments: $${{\Gamma }}_{1}=1.6\times \left[\begin{array}{ccc}\hfill \mathrm{cos}\,\theta \hfill & \hfill \mathrm{sin}\,\theta \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right],$$ $${{\Gamma }}_{2}=\left[\begin{array}{ccc}\hfill \mathrm{cos}\,\theta \hfill & \hfill \mathrm{sin}\,\theta \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right],$$ $$\eta =1,\quad \mu =1,\quad \rho =1,\quad \lambda =18$$

We set the threshold of the WMR velocity saturation constraint conditions in the system to 0 < v ≤ 5, and 0 ≤ w ≤ 5 respectively. The simulation results of WMR trajectory tracking are shown in Figures 3–7.

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

The tracking trajectories are shown in Figure 3 for different iterations. As the iterations increased, the motion trajectories of the WMR were basically repeated with the desired trajectories. As shown in Figure 4, the data-driven iterative learning double-loop control strategy not only accurately tracked the trajectory of the WMR, but also solved the problem of velocity jump during the tracking process. Additionally, because of the velocity saturation constraint of the WMR, the angular and linear velocities maintained threshold tracking at 70 and 150 iterations. Therefore, the velocity saturation constraint substantially affected the output of the system. However, when we increased the number of iterations to 300, the real velocity almost exactly matched the given velocity. Figure 5 shows the reference velocity tracking curve without velocity saturation constraint, through comparison, we found that the velocity saturation constraint did not influence the tracking performance of the data-driven iterative learning control scheme. Figure 6 shows that the WMR velocity profiles in the system are within the saturation constraints and have good performance. Figure 7 shows the control inputs for the WMR.

Figures 8 and 9 show the maximum error of the iterative process of trajectory and velocity tracking, respectively. We found, more intuitively, that the tracking error decreased as the number of iterations increased. The maximum tracking error in this iteration was the difference between the velocity threshold and the given velocity, and tended to be stable after 250 iterations.

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

We have compared the convergence speed under different values of λ, as shown in Figures 10 and 11. As seen from the figures, both parameters are able to track the desired trajectory perfectly, however, the smaller the parameter λ, the faster the system response and the faster the convergence rate.

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

CONCLUSIONS

In this study, we examined the WMR trajectory tracking problem with velocity saturation constraints. We designed a new double loop-control scheme with constraints. We fully considered the influence of the mechanical characteristics of the system on the trajectory tracking control performance. We used data-driven iterative learning control to design the outer- and inner-loop trajectory tracking control schemes for kinematic and dynamic models, respectively. The scheme uses only the I/O data of the dynamic model, the output velocity of the WMR is corrected online, and an effective control method is provided for the precise tracking of the trajectory of a WMR. The proposed method also solves the problem of velocity jump experienced by traditional control algorithms, and proves that the velocity saturation constraint does not affect the tracking performance of the data-driven iterative learning double-loop control scheme. The simulation results showed that the proposed strategy has high tracking accuracy and stable tracking capability.

ACKNOWLEDGEMENTS

This work is supported by the Innovation Project of Guangxi Graduate Education (Grant No. YCSW2022436).

CONFLICT OF INTEREST STATEMENT

No potential conflict of interest was reported by the authors.

DATA AVAILABILITY STATEMENT

Data available on request from the authors.

References

Kramer, J. , Scheutz, M. : Development environments for autonomous mobile robots: a survey. Aut. Robots 22(2), 101–132 (2007). [DOI: https://dx.doi.org/10.1007/s10514-006-9013-8]

Hou, Z.S. , Dong, H.R. , Jin, S.T. : Model‐free adaptive control with coordinates compensation for automatic car parking systems. Acta Autom. Sin. 41(4), 823–831 (2015)