Design of a counterbalance forklift based on a predictive anti-tip-over controller

1. Introduction

It is well known that industrial vehicles are one of the key elements in an industry. Forklifts are part of the necessary equipment for industries that may require material handling machinery. They allow the transportation of almost any kind of material thanks to the standardization of the platform called pallet. The problem to be studied in this paper is focused on the improvement of the forklifts, in particular a counterbalance forklift, a highly unstable vehicle in relation to roll and pith angles.

This kind of vehicles can lift the load up to considerable heights and therefore the height of its center of gravity depends on the working conditions, producing in some cases instability conditions and a significant risk of accidents. The main problem in the use of forklifts is their stability, especially while working under load. The most critical situations occur when the forklift is moving on uneven surfaces, at the beginning and the end of lifting or lowering the load, maneuvering the forklift, and accelerating or braking suddenly. This inherent instability causes a large number of accidents with loss of load, and damage and injuries of forklift operators.

Concerns about forklifts’ safety have been always present. Stout-Wiegand [28] presents a first study quantifying the number and importance of accidents where forklifts are involved. Bostelman [8] indicates that OSHA (Occupational Safety and Health Administration) in USA estimates that there are 110,000 accidents involving forklifts each year, approximately 31,600 employees suffer some type of injury. Classifying causes of lift truck accidents, 25.3% are tip-overs of forklifts.

Operational safety can be ensured in facilities with industrial vehicles only when safe practices are clearly established and carefully monitored. In facilities where autonomous vehicles are used, control systems must assure that accidents and incidents are prevented, and the vehicles must be equipped with safety devices. Bostelman and Shackleford [7] presents a research about advanced three-dimensional imaging sensors and their use in manufacturing towards improving forklift safety. Experiments are presented in this paper and show how the sensors can augment a forklift operator’s perception of obstacles nearby.

One of the main problems that appears and produces accidents is the inherent and characteristic instability of the forklift trucks. De Ninno and Uherka [10] developed a first initial study of forklift truck stability, establishing basic methods to obtain the equations to determine the critical slope for a vehicle.

Cheema and Sepehri [9] present a study on how the loading, wheelbase size, vehicle speed, top-heaviness and inclination affect the stability of forklifts with the goal to bring more insight into the stability of forklifts.

Simion et al. [27] present an equivalent dynamical model of the forklift system under different working conditions like as descending on a slope braking of the vehicle in translational motion and acceleration of the fork while lifting the load.

Lemerle et al. [16] analyze the vehicle dynamics and stability of four-wheeled forklift trucks in cornering situations, performing a parametrical study, examining the influence of certain important technical properties of the truck such as the maximum speed, the position of center of gravity, rear axle design features and tire properties.

Rinchi et al. [24] propose the implementation of sensor-based monitoring and control/supervision systems. It also presents a control system that is able to evaluate vehicle load conditions and to prevent unstable working conditions by limiting traction/braking performances.

Railsback et al. [21] examine the accelerations during the operation of stand-up forklifts, presenting experimental data acquired during stand-up forklift testing, and examining performance required for consensus and industry standards, industry practice and the safety implications of stand-up forklift acceleration.

Kichkin and Kichkina [14] discuss how to design a system to establish the stability of a forklift working with different weights using fuzzy logic and information technologies, oriented to help to the driver, but not for an autonomous vehicle.

Different authors have studied the problem of tip-over. Rebelle [20, 21]presents a numerical model for predicting forklifts truck tip-over. The measurements recorded during a test involving lateral partial tip-over were used to define a reference for dangerous driving situation. Based on these specific driving conditions, simulations were conducted by varying not only the forklift truck design and driving parameters, but also parameters relating to its tires. The results show that the forward velocity of the forklift truck, its center of gravity height, steering angle amplitude and the machine’s lateral wheel base are the most sensitive parameters amongst those studied.

Lambert [15] analyzes the problem of braking under load in a front load forklift. The results are clear: when travelling with the top of the forks 200 mm above the ground, on average, with just the 27% of the rated load, full braking would cause forward tip-over.

Due to the large number of accidents involving forklift trucks, there are numerous studies that focus on this problem. Most of them deal with conventional forklifts, not with Automated Guided Vehicles (AGV), and they are based on limiting the human factor. Typically, they limit the maximum speed, the acceleration capacity and the braking force. The first option consists in limiting the forklift performance, so it is stable under the most adverse conditions, but the problem is that in any other condition the truck is been underused. A more advanced solution consists in measuring the load in each axle and limiting the performance based on an analytical model [29]. Other controllers do not rely on such a rigid model and implement fuzzy logic to adapt the limit to the load variability [14].

This work presents a new concept in the field of forklifts stability. It consists on the introduction of a pitch control in the driving axle, which is the main contribution of tis paper. This concept has not been used until now in forklifts.

The initial step that motivated this work was originated from the idea of a novel industrial application of CAE starting from a new computational paradigm from classrooms to reality. The basic idea is that student must look for a real and interesting problem. Then he must analyze it and propose a solution, stating from a complete problem analysis and a model of the problem. After that, applying an innovative procedure, a scaled low-cost prototype must be built, respecting the dynamic characteristics of the real model, being this model used to validate a first conceptual idea of how to solve the problem. Following this scheme, a fist design and prototype of anti-tip-over control based on a PID pitch control was developed [5], serving to validate the feasibility of the conceptual idea as a first step.

Following this basic idea, in this paper, a more complex and sophisticated control system is developed, based on multi-objective and multivariable optimization methods.

In the field of optimization of a controller for airplanes, Rostami and Neri [25] propose an algorithm to address a multi-objective optimization problem. The proposed algorithm is an evolutionary algorithm based on the Covariance Matrix Adaptation Pareto Archived Evolution Strategy (CMA-PAES). The proposed algorithm has been tested in a seven-objective real-world application, i.e. the design of an aircraft lateral control system.

In the same field, Rostami et al. [26] present a novel algorithm for addressing multi-objective optimization problems, by employing a progressive preference articulation approach to decision making, derived from a statistical technique.

In the field of wheeled mobile robots Wang et al. [31] propose a novel trajectory tracking control approach for nonholonomic wheeled mobile robots where a proportional-integral-derivative based membrane controller is introduced to design the dynamic controller of wheeled mobile robots.

Because of the vehicle is automatically controlled, in the case of AGVs the moving direction is not predetermined. For this kind of vehicles, the trajectory is unknown and the future desired trajectory is known only over a finite horizon at each time step, following the spirit of Model Predictive Control (MPC).

The MPC technique [11, 17] is an advanced control technique for difficult multivariable control problems, widely used in the last developments for control of autonomous vehicles. Some applications of this control technique for autonomous vehicles are seen [4, 6, 20, 32].

The control used in this paper presents a MPC based on successive linearization of the non-linear vehicle model. The idea of using time varying models has been properly formalized recently. The work in [12, 13] is closest to the approach presented in this paper, by using an MPC scheme which has been successfully validated on a Boeing aircraft.

Following this approach, the anti-tip-over control developed in this paper will consider MPC, controlling the system while satisfying inequality constraints on the input and output variables.

The objective is to control the pitch vehicle dynamics via active control angle, the control input is the driving force and the goal is to follow the desired profile velocity as close as possible while fulfilling various constraints reflecting vehicle physical limits and design requirements. With these principles, a MPC control for the pitch angle has been designed.

The last step is the use of a new innovative concept for validation. This validation procedure is based on the developing of a scaled fast prototype. In this way, the simulation model is validated in a fast and economic way and errors detected at this early stage are solved saving a great amount of resources. This validation technique has been introduced by several authors [1, 3] and the cost reduction has been successfully proven. Scaling of mechatronic prototypes is subjected to some scaling rules and criteria, and in reference [2] an application of prototype scaling for railway vehicles is included and the criteria used there are discussed and extended to this work.

In this paper, Section 2 presents the description of the problem under study.

In Section 3, a simulation model is developed, first in order to understand the problems and the system behavior, and second to be the base for implementing this control. Section 3 presents the developed model and includes a stability analysis in different situations.

Section 4 presents the open loop simulation, showing several results corresponding to cases in which the system is stable, but also when the system is unstable and the tip-over is reached.

Section 5 includes the development of the designed MPC controller showing its results, demonstrating the feasibility of this control for this kind of vehicles.

Section 6 describes the dynamically scaled prototype manufactured to implement the controller and the criteria used in this paper, including validation results of the developed control. Finally, Section 7 includes the conclusions of this work.

2. Problem description

This paper studies an automated guided, counterbalance forklift. As it is automatically controlled, it does not depend on the human factor and we can improve the stability with the appropriate controller. The focus of this work is on improving forklifts’ stability and, at the same time, reducing the transportation times with the correspondent operating cost saving, analyzing vehicle stability and implementing a stability control.

The process of acceleration or deceleration when the fork is loaded, and the load is in its highest position, is one of the most critical situations regarding stability. A conventional forklift usually moves forward with the load in the front part, so the operator can control the load visually. However, in the case of AGVs the moving direction is not predetermined, it can move with the load either in the front or the rear part of the truck. This is hereinafter referred as front load forklift and rear load forklift, respectively.

During the acceleration process with the load lifted, the inertia forces can cause the forklift to tip-over if it is a rear load forklift (Fig. 1b).

However, in the case of a front load forklift, the torque produced by the inertia forces are compensated with the reaction in the caster wheel, so it cannot cause the forklift to tip over (Fig. 1a).

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During the braking process the situation is just the opposite. If it is a front load forklift, the inertia forces can cause the truck to tip-over, while if it is a rear load forklift, the torque produced by the inertia forces are compensated with the reaction in the caster wheel.

From the previous analysis, it is concluded that, if the forklift pitch angle reaches a limit value, the system becomes unstable, and the tip-over starts.

The open loop simulation is used to determine the dynamic conditions that produce this phenomenon of instability.

The longitudinal tip-over limit is defined as the maximum acceleration that the forklift may achieve without tipping over. In a first approach, this limit is theoretically calculated assuming that the tires are completely rigid.

The equilibrium of moments in B results as:

R A ⁢ ( A + B ) + M 1 ⁢ a ⁢ H + M 2 ⁢ a ⁢ h - M 1 ⁢ g ⁢ B + M 2 ⁢ g ⁢ l + T = 0(30)

In the limit, the vertical reaction in the front axle, R A , is zero. At this moment, the torque applied in the rear wheels’ axle, considering null the wheels’ inertia and roll resistance, is:

T lim = r ⁢ ( M 1 + M 2 ) ⁢ a lim(31)

Substituting Eq. (31) into Eq. (4) and considering R A = 0 , the longitudinal tip-over acceleration limit results:

a lim = M 1 ⁢ g ⁢ B - M 2 ⁢ g ⁢ l M 1 ⁢ H + M 2 ⁢ h + r ⁢ ( M 1 + M 2 )(32)

For the parameters considered in the model, the acceleration limit results as a lim = 0.695 m/s 2 .

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This implies that if the acceleration limit is exceeded and the pitch angle φ lim is reached, the forklift will tip-over. This effect is visualized in the open loop simulations carried out.

Figure 11 shows the results of a simulation with a driving force with acceleration under the acceleration limit, observing that the vehicle remains stable.

However, Fig. 12 shows the results of a simulation with a higher traction force, where the acceleration limit is exceeded. Then the system becomes unstable and the forklift overturns.

With the aim of increasing the forklift stability and to guaranty its safety, an anti-tip-over control has been designed considering model predictive control (MPC).

Usually, MPC control predicts future behavior using a linear time-varying (LTV) dynamic model. In many applications, this approach is sufficient for robust controller performance.

But in this case, because of the plant is strongly nonlinear and its characteristics vary dramatically with time when the front wheel loses its contact with the ground, an Adaptive MPC must be used by adapting the prediction model for changing operating conditions, by allowing the model parameters to evolve with time. A terminal constraint has been added in order to guaranty robustness.

The control used presents a MPC controller based on successive linearization of the non-linear vehicle model presented in Eq. (6). This non-linear model is linearized around the current operating point at each time step, and a linear MPC controller is designed for the resulting LTV system.

At each control interval, the adaptive MPC controller updates the plant model and the nominal conditions. Then, for the prediction model, the discrete-time states used can vary with time, which can be obtained from the nonlinear system Eq. (6).

The most frequent approach used in the literature is the explicit Euler method Eq. (33).

X t + 1 = X t + f t ⁢ Δ ⁢ t(33)

This method is easy to implement, but it requires a very small time-step Δ ⁢ t in order to obtain a good and numerically stable solution. Instead of it, and because of the characteristics of this model (nonlinear and highly stiff), the results can be improved, in terms of performance, if an implicit Euler method is used Eq. (34).

X t + 1 = X t + f t + 1 ⁢ Δ ⁢ t(34)

The implicit Euler method is suitable to keep the error in the result bounded. In this case, to achieve a given accuracy, it takes significantly less computational time to use an implicit method with larger time steps, even taking into account that an equation of the form Eq. (34) needs to be solved at each time step. This equation is solved by the fixed-point iteration method. Implications of the use of each one of this approach will be discussed in the section of simulation results.

The expression of f t is obtained through linearization at the operation point therefore from Eq. (8).

Taking into account that D u t = [ 0 ⁢ 0 ⁢ 0 ] T , we obtain:

X t + 1 = X t + ( X ˙ 0 + A t + 1 ( X t + 1 - X 0 ) + B t + 1 ( U - U 0 ) ) Δ t(35) Y t = C t ⁢ X t

where: t is the current control interval time index. X are the plant model states. In this case X = ( u v ω x y φ ) T . Y are the measured plant outputs. X = ( u ⁢ y A ⁢ φ ) T . Δ ⁢ t is the discrete time interval.

The objective of the control system is to track the velocity profile u 𝑟𝑒𝑓 maintaining the equilibrium position, therefore:

lim t → ∞ ⁡ ∥ u ⁢ ( t ) - u 𝑟𝑒𝑓 ⁢ ( t ) ∥ = 0 lim t → ∞ ⁡ ∥ y A ⁢ ( t ) - y A ⁢ ( 0 ) ∥ = 0(36) lim t → ∞ ⁡ ∥ φ ⁢ ( t ) - φ ⁢ ( 0 ) ∥ = 0

For the formulation of the MPC a prediction horizon N p is considered.

The notation X t + k | t represents the state vector at time t + k , predicted at time t , obtained by starting from the current state X t | t = x ⁢ ( t )

Over the prediction horizon [ t , t + N p ] , we define:

Y k | t as the predicted output trajectory,

U k | t as the predicted control input.

The cost function J ⁢ ( Y , U ) considered has the following expression:

min U 0 | t p , … , U N p - 1 | t p ⁡ J ⁢ ( Y , U ) = ∑ k = 0 N p - 1 J k ⁢ ( Y k | t , U k | t )(37)

subject to:

X k + 1 | t = X k | t + Δ ⁢ t ⁢ f k + 1 | t ⁢ ( X k + 1 | t ⁢ U )(38a) Y k | t = C k | t ⁢ X k | t(38b) X 0 | t = X t(38c) φ k | t + 1 ⩽ φ lim(38d) - F max ⩽ U k | t ⩽ F max(38e) - P max ⩽ u k | t ⋅ U k | t ⩽ P max(38f) y N p | t + X ⁢ A ⋅ φ N p | t - Y ⁢ A ⩽ 0(38g)

where [ U 0 | t , … , U N p - 1 | t ] denotes the unknown variables to be optimized.

Equation (38d) represents the maximum pitch angle limitation.

Constraint Eq. (38e) corresponds to the maximum allowed driving/braking force. Equation (38f) represents the maximum power constraint.

The way of ensuring stability is to add a terminal constraint which forces the state to take a particular value at the end of the predictive horizon as shown in [18, 19]. The Eq. (38g) corresponds to the terminal constraint used. This constraint is established to enforce that the vertical displacement for the point A is always negative and, consequently, the tip-over is not produced at the end of the predictive horizon. This terminal constraint ensures the stability of the proposed controller, because it ensures that the forklift is always in a stable position.

The function J k in Eq. (5) is the cost defined as:

J k ⁢ ( Y k | t , U k | t ) = J k ⁢ Y ⁢ ( Y k | t ) + J k ⁢ U ⁢ ( U k | t ) + J k ⁢ φ ⁢ l ⁢ ( U k | t ) + J k ⁢ a ⁢ ( Y k | t )(39)

where: J k ⁢ Y ⁢ ( Y k | t ) represents the cost of the output reference tracking, meaning that the controller prefers to follow the desired trajectory. J k ⁢ U ⁢ ( U k | t ) represents the cost of the control input tracking, meaning that the controller prefers to maintain a trajectory without acceleration. J k ⁢ φ ⁢ l ⁢ ( Y k | t ) represents the cost of the pitch angle limitation constraint, meaning that the controller tries to maintain always φ ⩽ φ lim . J k ⁢ a ⁢ ( Y k | t ) represents the cost of the terminal constraint, meaning that the controller tries to maintain always y A ⩽ 0 , maintaining always the point A in contact with the ground.

The full expressions for each component of the cost function are:

J k ⁢ Y ( Y k | t ) = K u ( u k | t - u 𝑟𝑒𝑓 ( k | t ) ) 2 + K y ( y A k | t - y A ( 0 ) ) 2 + K φ ( φ k | t - φ ( 0 ) ) 2(40) J k ⁢ U ⁢ ( U k | t ) = K U ⁢ ( F D ⁢ R k | t ) 2(41) J k ⁢ φ ⁢ l ⁢ ( Y k | t ) = K φ ⁢ l ⁢ ( φ k | t - φ lim ) 2(42) J k ⁢ a ⁢ ( Y N p | t ) = K a ⁢ ( y N p | t + X ⁢ A ⋅ φ N p | t - Y ⁢ A ) 2(43)

being: u 𝑟𝑒𝑓 the desired speed profile, y A ⁢ ( 0 ) and φ ⁢ ( 0 ) the vertical displacement of the pitch angle at the static equilibrium position.

For the simulation of the proposed model, a MatLab code has been develope. This code consists of a constrained non-linear optimization problem, that has been developed by using the interface Yalmip [17] and Ipopt [30], solver used for nonlinear optimization problems.

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A value of Δ ⁢ t = 0.01 s has been used because is small enough to guaranty stability, however if an explicit Euler method has been used, the required Δ ⁢ t would be 0.0015 s. Only 2 iterations have been considered for the implicit Euler method, being enough to ensure the numerical stability.

The predictive horizon N p is set to be 10 as a compromise between computation time and feasibility of the solution.

Table 1 lists the corresponding weights in the cost function.

In the simulation, the box constraint on tracking/ braking force is reflected by the maximum driving and braking force F max , limited also by the maximum power P max = 4500 w.

The a lim obtained from Eq. (32) is 0.695 ⁢ m / s 2 , that corresponds to a maximum force of F lim = 2562 N. and φ lim = 0.00385 rad.

First, in order to check the stability, a simulation has been done defining a perturbation by means of a horizontal force defined as a pulse from 1.00 s to 1.02 s with different amplitudes. Figure 13 shows that the system is stable, and it also respects the constraints.

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For the simulations, a speed profile is stablished, as shown in Fig. 14 in order to study the system behavior. This speed profile is used to test an acceleration maneuver and, after that, a braking maneuver.

This profile means that when the simulation reaches t = 1 s, the vehicle tries to reach a speed of 3.5 m/s as soon as possible, depending on the acceleration produced by its driving force. And when simulation reaches t = 8 s, the vehicle starts braking until the vehicle stops, by applying the maximum braking force. In both braking and driving manoeuver, the maximum power is also considered as a constraint.

Three different cases for F max are considered. The first one considers F max = 1800 N. In this case, and because of the acceleration limit is a lim = 0.695 m/s 2 , this force produces an acceleration under the limit, and consequently, the vehicle does not tip-over, maintaining its stability.

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In the second one, F max = 2400 N. This case is just on the limit of stability. When the vehicle starts the acceleration manoeuver, this force produces an acceleration above the limit in the transient period, but the controller maintains the stability.

And in the third case, F max = 4800 N, produces an acceleration above the limit, and consequently, the vehicle is unstable if no anti tip-over control is applied.

Figures 15 to 21 show the results for these simulations. In these figures, several variables are represented, including simultaneously in the same plot the results for each F max

Figures 15 to 18 represent the control input (driving/braking force) and the pitch angle. In the first part of the simulation, when the vehicle accelerates, it is observed that for F max = 1800 N, the driving force applied reaches the maximum during the maneuver. For F max = 2400 N, the driving force is reduced several seconds at the beginning of the acceleration manoeuver, in order to prevent the instability that appears in the transient but remains at the maximum power in the rest of the simulation, because of a lim is not reached.

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For F max = 4800 N, the controller acts in the transient, and also the driving force is limited to a value of F lim = 2562 N, due to the fact that F max is higher than the maximum under the stability limit. In the rest of the simulation, the controller acts only following the velocity profile, but no limiting the force.

On the other hand, the pitch angle (Figs 17 and 18) is not affected in the case of F max = 1800 N because of it is always under the stability limit, but in the second and third case, the pitch angle has a limitation imposed by the controller.

In the case of F max = 2400 N, a limitation is imposed due to the fact that during transient period of the acceleration manoeuver, the φ lim is reached.

And in the case of F max = 4800 N, the driving force is limited to F lim = 2562 N, because the system would be unstable without controller.

Figure 19 shows the vertical displacements y A and y B for the contact points of the front and rear wheels. Following the sign convention used, a negative displacement represents the case where the tire is compressed and in contact with the ground, and a positive value means that the tire loses its contact with the ground.

The stability control imposes that y A must be always less or equal to 0, as shown in Fig. 19.

From an operational point of view, Figs 20 and 21 must be analyzed.

Figure 20 shows the velocity followed in each simulation, showing that for F max = 2400 N and F max = 4800 N, the maximum velocity of 3.5 m/s is reached, but if F max = 1800 N, the driving force is not enough to reach the maximum velocity of 3.5 m/s in the interval simulated.

Figure 20 shows that, in the acceleration phase, for the cases of F max = 2400 N and F max = 4800 N the velocity increases in a similar way, and no significant differences are appreciated for the different cases, because of both of them are close or in the limit of allowed acceleration.

But in the braking phase, significant differences appear, because in this case, the vehicle does not have stability problems, and the braking force is not limited.

So, as a conclusion, the stability control improves the operation of the forklift because of it uses the maximum limit force without stability problems when accelerating and it uses the maximum available force when braking, being the total time to carry out the whole maneuver lower for higher driving/brakingforces, as shown in Fig. 21.

This paper presents the design of an anti-tip-over control applied to a counterbalance forklift, a highly unstable vehicle, mainly in relation with its pith angle.

With this new concept in the field of forklifts, stability is significantly improved, since this control not only decreases the risk of accidents, but also reduces the transportation times and consequently the operational costs.

First, an analytical study has been developed in order to analyze the characteristics of this kind of vehicles and their stability problems. As a result of this analytical study, the conditions for stability are obtained showing when this type of vehicle becomes instable.

After that, a MPC controller is designed and implemented. This kind of control has been successfully used in autonomous vehicles, being the objective of this paper to demonstrate how feasible it is in this particular vehicle.

Several simulations and test are carried out. The results are very satisfactory, showing that the forklift is able to maintain its stability under conditions that a vehicle without control would be unstable. In particular, in acceleration, the vehicle is able to avoid the tip-over by controlling the pitch angle, but when braking, the forklift is able to use all the braking capacity. Thus, as a conclusion, the vehicle with the controller is able to carry out acceleration and braking manoeuvers in a more quick and safe way.

In order to validate the control strategy and ensure that the controller is feasible, an innovative procedure for validation using a scaled prototype has been developed allowing to detect design problems in a previous stage before to build the real and high cost prototype. Simulation have been replicated with experimental tests obtaining equivalent results.

Hence, and because of all the simulations and test have obtained positive results, it is concluded that the MPC controller, based on the pitch angle regulation, increases safety, significantly reduces the transportation times, and consequently reduces the operational costs, being thus especially advantageous.