A Model-based Parameter Optimization Control Strategy for Trajectory Tracking with Torque and Velocity Constraints

The wheeled robots trajectory tracking control methods rarely constrain the torque and speed at the same time. In actual application, the torque and speed of the robot cannot exceed the saturation limit of the actuator. This paper develops a model-based trajectory tracking parameter optimization controller with both velocity and torque constraints, using a gradient descent parameter iterative learning strategy to minimize the settling time index of the system. Trajectory tracking time optimization methods usually require a given analytical expression of the system time, while this time optimization method only requires that the settling time is solvable. The MATLAB simulation experiments show that the proposed parameter optimization controller for trajectory tracking can perform velocity and torque constraints while having a relatively good overall rapidity time index. If the resolution of the robot sensor can meet the design requirements, the optimization method can strictly control the system torque maximum to a reasonably small expected value. When the resolution of the robot sensor is limited, this optimization method can restrict the system torque maximum within a reasonable saturation constraint range.


Introduction
The robot trajectory tracking control has become a hot field of robot research [1]- [6]. Scientists have studied various wheeled robots controller design methods [7]- [10]. When designing the trajectory tracking controller, the saturation constraint of the actuator must be considered [11]- [22]. In the existing research on trajectory tracking control, there are many papers on the velocity constraint control of the actuator [11]- [16], and a few papers on the torque constraint control [17]- [18]. Both constraints are rarely considered at the same time [19].

Robot trajectory tracking with velocity constraints
In practice, the operating technical specifications of motors are always limited. High angular or linear velocity may lead to the robot losing stability and rollover or exceeding the safe travel range to cause a collision or accident. Therefore, it is necessary to control the speed limit of the robot [11]- [16].
Yu et al. [11] used the position error vector to construct the Lyapunov function and designed a trajectory tracking control law for a nonholonomic unicycle vehicle to realize the position vector errorless tracking with speed limitations. The simulation illustrated that tuning the controller parameters can realize limiting the velocity constraints.
Liu et al. [12] designed a controller with velocity constraints allowing the unicycle formation to track the reference trajectory, and the steady-state error converges to zero. The leader and follower of the system are both subjected to linear and angular velocity constraints due to the physical limitation. The experiment results proved that the velocity commands would not exceed the physical restraints.
Yang et al. [13] employed a model predictive control to a nonholonomic wheeled mobile robot for trajectory tracking in the presence of velocity constraints and other softening constraints. The presented control strategy contained a feedforward controller and a feedback controller to achieve trajectory tracking smoothly and reach the requirements of the softening constraints. For improving real-time robustness, a linearised tracking error model was utilized to predict system behaviors.
Guo et al. [14] took the linear and angular velocity limits into account to design a unicycle-type robot trajectory tracking controller. The tracking experiments illustrated that the proposed controller has strong robustness and stability. The controller tracked a variety of different trajectories accurately.
Xu et al. [15] presented a tracking control strategy integrating a biologically inspired backstepping controller and a torque controller with an unscented Kalman filter and Kalman filter to avoid and reduce the velocity jumps and overshoots and provide smooth velocity commands. The upper and lower bounds of the speed are set by the B and D parameters in the biologically inspired neural network, and the control constraints are realized. The proposed method can track the desired trajectory more accurately, provide smooth velocity commands and reduce the effects of the noises.
Chen et al. [16] devised a model-based controller and an adaptive neural network controller for a wheeled mobile robot with velocity constraints. This paper has accomplished two control law devisals, stability analysis, simulation, and practical experiments. The model controller will track the required trajectory well and limit the speed to meet the saturation constraint. The model-based control method enables the position error to converge to a small neighborhood of zero without violating the speed constraint. And the adaptive neural network is suitable for uncertain robot systems to approximate uncertain or time-varying parameters and improve the control robustness.

Robot trajectory tracking with torque constraints
The torque is physically bounded by the practical power supply limitation. Excessive torque causes saturation of the actuator, and it may deteriorate the tracking performance of the controller, especially in transient response. A long-term torque saturation may cause thermal, electrical, or mechanical failure of the actuator. Accordingly, controlling the torque to make the actu-ator not saturated has attracted the attention of the robot field [17]- [18].
Ailon et al. [17] addressed the trajectory tracking control of differentially driven nonholonomic wheeled mobile robots subjecting to bounded torque inputs. In this article, the applied torques are bounded for a known constant. Under the constraint conditions, the controller can ensure accurate tracking and guarantee that the vehicle does not deviate from the trajectory. The presented control algorithm applied hyperbolic functions for tracking a reference path with an associated timing law under the applied forces and torques constraints.
Wang et al. [18] proposed a model predictive control and Extended State Observer-based Adaptive Sliding Mode Control (ESO-AMSC) for formation control of Mecanum-Wheeled Mobile Robots (MWMR) with physical constraints and model uncertainties. The designed adaptive sliding mode controller with extended state observer has torque saturation constraints. The simulation results show that the linear model predictive controller can deal with the kinematic constraints of MWMRs, and the ESO-ASMC can address the input torque saturation constraints and remove the effect of model uncertainties and unknown disturbances.

Main contributions
As mentioned above, the wheeled robot should restrict the torque and speed. These two constraints need to be considered at one time to make the designed robot control system operate safely [19].
Yue et al. [19] designed a robust model predictive control strategy to handle the trajectory tracking for an underactuated two-wheeled inverted pendulum vehicle with various physical constraints. The authors treat the tilt angle, angular velocity, longitudinal and rotational travel speeds, and input torques as the physical constraints from outside the system. And they employed a Lyapunov candidate to handle the physical limitations from outside. Simultaneously, they exploited the model predictive control to solve the diversity of the physical constraints and unavoidable disturbances on the inner-loop system. The numerical simulations verified that the controller could track the designated trajectory with various physical limits and interferences.
Compared with the control method proposed in [11]- [15], [17], [19], which can not achieve the error-free tracking well or the settling time is too long, the model-based control method presented in [16] can quickly realize the error-free track of the position vector, and the control method is simple. The controller designed in [18] only carried out a comparative discussion of several different Am o d e l -b a s e dp a r a m e t e ro p t i m i z a t i o nc o n t r o ls t r a t e g yf o rt r aj e c t o r yt r a c k i n gw i t ht o r q u ea n dv e l o c i t yc o n s t r a i n t s 3 control methods and did not provide the experimental curve diagram and data of the position vector tracking of the reference trajectory.
This paper develops a model-based parameter optimization scheme for the speed and torque constraints. The general time optimization method needs an analytical time expression [20]- [28]. The model-based parameter optimal method proposed in this paper only needs the settling time to be calculatable. Here we take the gradient descent optimal method to design the modelbased parameter optimal strategy for settling time [29]- [31]. With the proposed method, the position and velocity errors can converge to zero. The main contributions of this paper are as follows: 1) A model-based parameter optimization trajectory tracking control scheme with velocity and torque constraints for wheeled mobile robots is proposed.
2) The model-based parameter optimization scheme proposed in this paper does not need to provide a given analytical time expression, and it only requires that the settling time is solvable.
3) This paper provides two optimization schemes. In the first scheme, the system torque maximum can be an accurate value when the sensor accuracy meets the requirements. And in the second scheme, the torque maximum can be in a saturation range when the sensor accuracy is limited. 4) In MATLAB simulation experiments, comparing with traditional backstepping control, fast terminal sliding mode control, and the model-based control [16], the parameter optimization control method presented in this paper is more effective to restrict the input torques and velocities with a reasonable settling time.
The rest of this paper is organized as follows. Section 2 introduces the mathematical model of the wheeled mobile robot. Section 3 discusses an improved modelbased controller with speed limitations. In Section 4, the model-based parameter optimization method with input torque and velocity limits is given. Section 5 studies the simulation experiments of three tracking trajectories by five control metohds. Section 6 is the discussion and comparison of the experimental results. Section 7 introduces the conclusions of this paper.

Wheeled mobile robots model
The designed differentially steered Wheeled Mobile Robot (WMR) is shown in Fig. 1, v v =[v, ω] T 2 R 2 , v and ω are the linear and angular velocities, respectively. x 2 R and y 2 R are the Cartesian coordinates of the mass center, and θ 2 R is the heading angle of the WMR shown in Fig. 1. There are some assumptions adopted in the WMR for the established trajectory tracking system model. To maintain the nonholonomic constraint, the WMR is not able to move sideways. The centroid of the WMR coincides with the geometric center [10]- [16].
In Fig. 1, r is the radius of the wheels, and the distance between the two rear wheels is 2l. m is the mass of the WMR, the moment of inertia is J. q = [x, y, θ] T 2 R 3 is the position vector [1]- [2].
In this paper, the physical actuator velocity constraints are defined as [11]: The desired trajectory position vector is where v d and ω d are desired velocity satisfying [11] ⇢ To meet the asymptotic stability of tracking the position vector q, the auxiliary velocity input is defined as [32] ⇢ where k 1 , k 2 , and k 3 are adjustable positive constants, and k 2 1.
In V 1 , 1 2 (v v ai ) 2 is used for decreasing k v v ai k, and 1 2 (ω ω ai ) 2 is employed to reduce k ω ω ai k, f 1 (v)i s utilized to constraint linear velocity v, f 2 (ω) is applied to limit angular velocity ω, and 1 2 (ω mp ω ai ) 2 f 2 (ω mp ) are used to ensure that the minimum value of V 1 is 0 [16].
The detailed proof process in the literature [16] demonstrated that using the control law (17) with the velocity constraints |v| <v cm1 , |ω| <ω cm1 , |v ai | <v cm2 , |ω ai | <ω cm2 , the velocity and position errors of the WMR system would converge to zero. The closed-loop control system is asymptotically stable, and the system states would not violate the constraints [16].
Supplement: The mian difference between this paper and the literature [16] is formula (29), helping the controller to achieve stably error-free tracking. The first control law in [16] referred to in this paper is the modelbased control for the system with known model and parameters. The second control law on the title of [16] not mentioned in this paper is for the system having an unknown model or parameter.

Model-based parameter optimization control
The improved model-based control method realize the error-free tracking of the position and velocity vectors quickly with the speed constraints. This paper wants to achieve the non-error track combined with the speed and torque constraints. Adding torque constraints to the system may prolong the settling time. Therefore, we addressed a model-based parameter optimization control scheme to decrease the increasing settling time by the torque constraints.
The gradient descent method updates iteratively along the negative gradient direction of the objective function and finally converges to the minimum point. Cauchy proposed the gradient descent method in 1847. Due to the simple concept, the gradient descent method has become one of the most commonly used parameter minimization methods [29]- [31]. The basic idea of this method is to assume that the gradient vector is in the steepest ascending direction and take a sufficiently small step in the opposite direction, which causes the function value to be lower than the previous value. Repeat this process to reach the minimum value of the function.
The continuous gradient descent function iṡ where, ε>0i st h es t e ps i z e .rf (x)r e p r e s e n t st h e gradient of f (x) at x. The corresponding discrete gradient descent function is where a is the number of iteration steps, and x(a)isthe a-th step variable value. The gradient descent method is an iterative optimization algorithm for obtaining an approximate solution from the next step of the current step until reaching the termination condition in a stepwise loop. From equation (16), we can conclude In the equations (1)- (29), the all adjustable parameters are k 1 ,k 2 ,k 3 ,ρ 1 ,ρ 2 . And they are all constants greater than zero. Define ζ =[k 1 ,k 2 ,k 3 ,ρ 1 ,ρ 2 ] T 2 (R + ) 5 ,

Algorithm 1 Model-based parameter optimization algorithm
at the time j +1. j = j +1. 8: Repeat steps 1-7, calculate the n-dimension output vector series on the time series t. 9: Update the parameter values k 1 ,k 2 ,k 3 ,ρ 1 ,ρ 2 by equations (34). 10: Repeat steps 1-9, using the equation (33) to get the smallest t s satisfying the input torques limits, where t s is calculated by the solved position vector sequence according to the t s definition of 2% error band. 11: return x e ,y e ,θ e ,v,ω,τ 1 ,τ 2 ,x,y,θ,t s ,a.
where k 2 1. The optimization goal is to minimize settling time t s , when the input torques are limited by |τ 1 |τ 1m , |τ 2 |τ 2m . t s1 ,t s2 ,t s3 are the settling times of the position errors x e ,y e ,θ e ,r e s p e c t i v e l y .D efine t s = max(t s1 ,t s2 ,t s3 ).
For keeping |τ 1 |τ 1m , |τ 2 |τ 2m , t s may increase to cause the position error series x e ,y e ,θ e to fail to converge to zero within the limited working time t.I n practical applications, we usually hope that the settling time t s is minimal.
Set the optimal objective function as The adjustable parameter optimization formula obtained by the gradient descent method is .
Since this strategy uses the model-based controller with speed limitation, the model-based parameter optimization method has both the speed and torque constraints, and its settling time can be the smallest. Furthermore, in Algorithm 1, the optimization strategy only needs the settling time to be solvable and does not need to have an analytic expression.

Simulations and experiments
In this section, we did the MATLAB simulation experiments to compare the backstepping control, the fast terminal sliding mode control, the improved model-based control, and the parameter optimization control scheme proposed in the paper to track different trajectories. The backstepping control, the fast terminal sliding mode control, and the model-based control methods are effective control methods for robots to achieve trajectory tracking. This section utilizes these three methods to carry out simulation comparison experiments to verify the effectiveness of the method proposed in this paper.

Spiral line trajectory tracking
In this part, spiral line trajectory tracking simulation experiments are carried out. The experiment data of Fig. 4-Fig. 5 are shown in Table 2. The desired velocities are given as v d =0.2+0.001t, ω d =0.1+0.008t.

Discussion
From Fig. 2-Fig. 7 and Table 1-Table 3, it can be observed that all the five control schemes can track the three desired trajectories, and the error signals can converge to a small neighborhood of zero. However, from  Fig. 2-Fig. 7 and Table 1- Table 3, we can observe the following.
(A). The latter three control methods can better limit the absolute maximum value of the speeds and torques compared with the first two traditional control methods. The maximum torques acquired by the first two classic control algorithms are too large, which leads to the large size and high cost of the motor.
(B). The latter two model-based parameter optimization schemes, compared with the improved modelbased control method, realize the optimization of the settling time and the overall rapidity. Moreover, the model-based parameter optimization schemes reduce the oscillation amplitude that appears in the improved modelbased control method and improves the riding index of the control system.
(C). The model-based parameter optimal scheme 1 can strictly limit the maximum absolute value of the torque to a small value (e.g. 2.000Nm in Table 1, 3.000Nm in Table 2, and 56.000Nm in Table 3) through the precise setting of the parameters. When the accuracy of the robot sensor meets the parameter setting requirements, this optimization scheme can achieve strict control of the absolute maximum value of the torque.
(D). The optimal scheme 2 also realizes the limitation of the speed, torque, and settling time, and its requirements on the sensor accuracy are much lower than that of optimal scheme 1. The adjustable parameters in optimal scheme 1 need to keep 15 digits after the decimal point, while the optimal scheme 2 only needs to keep 1 digit after the decimal point. Compared with optimal scheme 1, optimal scheme 2 can be widely applicable to more environments and has lower application costs.
The experiment data of the circle, spiral line, and parabolic line trajectories in Table 1-3 are averaged to obtain comparison histograms shown in Fig. 8-Fig. 10. The data in Table 1-3 are the absolute maximum values of the indexes. Averaging the data in Table 1-3   ries are given in Fig. 9. Fig. 10 displays the comparison of the absolute maximum mean value of torques of three trajectories. The following conclusions can be obtained from the analysis of the three comparison histograms.
(1). The position error absolute maximum mean values of the improved model-based control algorithm are the largest. The absolute maximum mean values of the position error of the other four algorithms only have little difference.
(2). The absolute maximum mean value of the linear velocity of the backstepping algorithm is the largest, and the absolute maximum mean value of the angular velocity of the sliding mode algorithm is the largest. The maximum absolute mean values of linear and angular velocities obtained by the model-based parameter optimization algorithms are relatively small.
(3). The maximum absolute mean value of the settling time of sliding mode control is the largest, and the maximum absolute mean values of the settling time of the optimization algorithms proposed in this paper are the smallest.
(4). The maximum absolute mean values of the torques of the backstepping method are the largest, and the maximum absolute mean values of the torques of the optimization algorithms proposed in this paper are the smallest.
Judging from the experimental curves and data of the torques, settling time, speed, and position error of  the five algorithms, the model-based parameter optimization algorithm proposed in this paper is the best. And the other three algorithms have their advantages and disadvantages under different indicators.

Conclusion
Regarding the above research of trajectory tracking, the control torque and velocity obtained based on the first two traditional classic trajectory tracking control algorithms may cause large initial values. We design a model-based parameter optimization algorithm with torque and velocity constraints. Constraining the torque and velocity makes this proposed method succeeds in settling down the problems of oversaturation, mechanical size constraints, and energy utilization efficiency caused by excessive torque and speed of the motor.
The limitation of the torque and velocity will cause the rapidity index to deteriorate. The model-based parameter optimization method introduced in this paper performs torque and velocity constraints while minimizing the weakening of the rapidity. This research method has achieved a satisfying balance between the size of the input torque and velocity values and the size of the settling time value, which has a great practical application value.
The majority of the time optimization methods need to provide an analytical expression of the system time, while this parameter time optimization strategy only needs that the system time can be solvable. Only needing the system time can be solvable makes this time optimization scheme easier to implement and solve.
Furthermore, the first three methods can only use the trial and error method or the empirical method to set the adjustable control parameters. The model-based parameter optimization method proposed in this paper presents design schemes and solutions to select adjustable control parameters through iterative learning based on parameter initial setting values.
In the research of this paper, we caught sight of that the model-based parameter optimization method sometimes traps in the local optima. The next step of our work plan is to seek a control method to solve the globally optimal value.