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Trajectory Tracking Control for a Container Transferring
Mobile Robot Based on an Improved NMPC
Xuehong Zhu1*,Fei Liu1,Lizhen Jia2,Ning Fan1
1.Tianjin Sino-German University of Applied Sciences, School of Mechanical
Engineering, 300350
2.Civil Aviation University of China, School of Transportation Science and
Engineering, 300300
**Corresponding Author:Xuehong Zhu
Email: zhuxuehong@tsguas.edu.cn
Abstract.Container Transferring Mobile Robots are widely used in container
handling, narrow-aisle transfer, workstation delivery, and automated loading and
unloading tasks. These robots usually operate in environments with dense shelving,
narrow passages, and clearly defined boundaries. During practical operation, they also
frequently perform start-stop motions, pickup tasks, and delivery tasks. These factors
make trajectory tracking more challenging, especially when the robot is affected by
motion-state switching, reference-path curvature variation, and limited aisle space. In
such conditions, attitude fluctuations and control instability may occur, increasing the
risk of collision between the robot body and the aisle boundaries.To address this
problem, this paper proposes a constrained trajectory-tracking control method for a
Container Transferring Mobile Robot based on a discretized model derived from
Lagrangian dynamics. First, the dynamic model of the robot is established using the
Lagrange method, and a prediction model is obtained through discretization. Then,
considering the vehicle width, aisle width, and safety-margin requirements, a virtual
safety corridor is constructed to describe the allowable motion range of the robot body.
This corridor is introduced into the trajectory-tracking optimization problem as a
geometric boundary constraint, so that the operating region of the robot can be limited
during prediction and control.The hard-constraint formulation can prevent the robot
from crossing the aisle boundary when the optimization problem remains feasible.
However, in cases with large initial deviations, rapid changes in reference-path
curvature, or further restricted aisle space, the feasible region of the optimization
problem may shrink significantly. This may lead to optimization infeasibility and affect
the continuous operation of the controller. To improve the applicability of the controller
under such conditions, nonnegative slack variables are introduced to soften the
geometric boundary constraints, and a linear penalty term is added to the cost function.
In addition, an adaptive penalty mechanism based on the degree of constraint violation
is designed to enhance constraint recovery near the boundary.Simulation and hardware
experimental results show that the proposed method maintains optimization feasibility
when the initial lateral deviation exceeds the prescribed corridor threshold. Compared
with the standard NMPC and the fixed-weight soft-constrained NMPC, the proposed
controller reduces the corridor-violation duration and improves boundary-recovery
Xuehong Zhu et al.
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performance, while maintaining acceptable tracking accuracy and real-time
computational performance.
Keywords. differential-drive mobile robot; container transferring mobile robot;
trajectory tracking control; model predictive control; constrained control.
Introduction
With rapid developments in intelligent manufacturing, container transferring mobile
robots are increasingly used in scenarios such as material handling, high-density
narrow-aisle transfer, and flexible production. Unlike navigation tasks in open
environments, these robots usually work in dense and narrow industrial spaces with
clear physical boundaries. They are required not only to track reference trajectories
accurately, but also to avoid collisions with shelves, containers, and surrounding
equipment. Klančar et al.[1]pointed out that mobile robots in industrial environments
must satisfy both path-tracking and safe-operation requirements within constrained
spaces. Achirei et al.[2]showed that trajectory control in logistics environments needs
to consider environmental constraints and online control performance at the same time.
Lucet et al.[3]studied autonomous forklift navigation in a cluttered logistics factory
and emphasized the importance of navigation corridors and tracking-control constraints
for safe operation in space-limited logistics environments. Therefore, accurate and safe
trajectory tracking in confined spaces remains an important control problem for
container transferring mobile robots.
For mobile robot trajectory tracking, controller design usually starts from either a
kinematic model or a dynamic model. In scenarios with rapid input changes and non-
negligible dynamic responses, a purely kinematic model is often insufficient to describe
the actual motion evolution of the robot. Dynamic models based on physical
mechanisms are therefore more suitable for high-precision trajectory-tracking control.
Zhang et al.[4]established a robot chassis dynamic model using the Lagrange equations
and considered factors such as the rotational inertia of the driving wheels, which
improved the description of the robot’s dynamic behavior. Moritz et al.[5]developed a
differential-drive robot model that includes dynamic modeling, kinematic modeling,
linearization, and stabilizing control design, providing a physical basis for constrained
controller design. Liu et al.[6]pointed out that the control design of wheeled mobile
robots should consider both kinematic and dynamic characteristics to handle
nonlinearity, underactuation, and constraints.
Model predictive control has become an important method for mobile robot trajectory
tracking because it can handle system dynamics and constraints within a receding-
horizon framework. Köhler et al.[7]noted that MPC is well suited for tracking problems
and constrained nonlinear systems. Wang et al.[8]applied nonlinear model predictive
control to pose tracking of skid-steered mobile robots and verified its effectiveness for
complex dynamic systems. Peng et al.[9]combined a nonlinear disturbance observer
with MPC to improve the control performance of wheeled mobile robots under complex
operating conditions. Tang et al.[10]pointed out that the kinematic and dynamic
characteristics of mobile robots make accurate modeling and effective control a
Trajectory Tracking Control for a Container Transferring Mobile Robot Based on an Improved
3
persistent challenge. Tang et al.[11]further proposed a geometric MPC method for
wheeled mobile robot trajectory tracking, showing that the description of robot
configuration and tracking error has an important influence on MPC tracking
performance.
However, in narrow-corridor container-handling scenarios, a dynamic model and a
standard NMPC framework alone are still not sufficient. In many existing studies,
spatial boundaries or safety conditions are directly imposed on the optimization
problem as hard constraints. Fnadi et al.[12]integrated steering and sideslip constraints
into a constrained model predictive control problem. Nam et al.[13]reinforced safety
boundaries during path tracking through independent constraints. Jian et
al.[14]combined dynamic control barrier functions with MPC for safety-critical
obstacle avoidance of mobile robots, further showing the importance of introducing
safety constraints into predictive control. These methods can strictly enforce boundary
and physical constraints when the optimization problem is feasible. However, when the
robot has a large initial deviation, the trajectory curvature changes rapidly, or the
available corridor space is limited, the feasible set within the prediction horizon may
shrink significantly. In severe cases, the optimization problem may become infeasible.
For container transferring mobile robots, lateral deviations may occur during pickup,
transfer, placement, and attitude adjustment. If the geometric boundary is always
treated as a hard constraint, the controller can provide strong boundary restriction, but
its feasibility may be weakened. Khan et al.[15]also pointed out that strict constraints
and complex operating conditions can affect the online continuity and practical
deployment of predictive control.
To reduce the infeasibility caused by hard constraints, soft-constrained MPC introduces
slack variables into the constraints so that the feasible set of the optimization problem
can be enlarged. Wabersich et al.[16]proposed a soft-constrained MPC formulation and
showed that the control problem can remain tractable even when constraint-violating
trajectories exist. Gracia et al.[17]further discussed the implementation of soft-
constrained MPC for tracking problems and emphasized its role in maintaining
optimization solvability. Liu et al.[18]developed an optimization-based local planner
for nonholonomic autonomous mobile robots in semi-structured environments, where
safety and feasibility were both considered in constrained navigation. These studies
suggest that soft constraints can alleviate the limitations of hard-constrained methods,
especially the problems of feasible-set shrinkage and optimization infeasibility.
Softened constraints have previously been introduced into MPC-based trajectory
tracking of wheeled mobile robots. For example, Yang et al.[23] applied MPC with
softening constraints to a nonholonomic wheeled mobile robot to avoid infeasibility
caused by strict constraints. Stochastic MPC with soft probability constraints has also
been investigated for wheeled mobile robot trajectory tracking[24]. However, the
formulation of a body-size-dependent virtual corridor and the recovery behavior from
initially infeasible corridor conditions have received comparatively limited attention in
torque-level NMPC for container transferring mobile robots.
Nevertheless, for the virtual-corridor recovery problem considered in this paper, a
conventional fixed quadratic slack penalty may still have limitations. This design can
reduce constraint violation, but when the robot moves near the boundary, the optimizer
Xuehong Zhu
4
may still retain a small nonzero slack value to improve local tracking accuracy or input
smoothness. As a result, slight boundary violations may remain during the steady-state
phase. For container transferring mobile robots operating in narrow corridors, such
small but persistent violations weaken the practical meaning of geometric safety
constraints. Therefore, it is necessary to further strengthen the compression of slack
variables while preserving optimization feasibility, so that the controller can recover
the original boundary constraint as much as possible after the robot returns to the
feasible region. Sun et al.[19]proposed a multi-constraint MPC method for forklift
trajectory control, indicating that stability and safety constraints need to be considered
together in industrial vehicle applications. Magalhães and Pimenta[20]studied
distributed MPC for safety-critical obstacle avoidance in multi-robot systems, further
showing that online solvability and safety constraints remain important issues in
complex constrained robotic systems.
This paper proposes an improved soft-constrained nonlinear model predictive control
method for trajectory tracking of container transferring mobile robots in narrow-
corridor environments. The method is based on a discretized Lagrangian dynamic
model. First, the dynamic model of the container transferring mobile robot is
established using the Lagrange method and then discretized to construct the prediction
model for receding-horizon optimization. Next, the vehicle width, corridor width, and
safety margin are used to convert the lateral tracking error into a virtual safety-corridor
constraint, which is embedded into the NMPC optimization problem as a geometric
safety boundary. Nonnegative slack variables are then introduced to soften the original
geometric hard constraints, improving controller feasibility under large initial
deviations and complex trajectory curvatures. To avoid insufficient slack compression
near the boundary, a linear penalty term is added to the slack-variable cost function,
and an adaptive penalty mechanism is designed according to the degree of constraint
violation. In this way, the proposed method limits excessive slack-variable growth and
helps the controller recover the boundary constraint after the robot re-enters the feasible
region of the original hard-constrained problem. While retaining the feasibility
advantage of soft constraints, the proposed method is intended to improve boundary-
recovery performance and reduce corridor-violation exposure, rather than to provide
an absolute geometric-safety guarantee.
1 Modeling and Analysis of a Differential-Drive Container
Transferring Mobile Robot
1.1 Kinematic Modeling
The differential-drive container transferring mobile robot is simplified as a differential-
drive mobile platform. As shown in Fig. 1.1,OXYis the global coordinate system, and
OPXPYPis the body-fixed coordinate system attached to the robot chassis. Under the
assumptions of pure rolling and no lateral slipping, the kinematic relationship of the
robot can be established.
Trajectory Tracking Control for a Container Transferring Mobile Robot Based on an Improved
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Fig. 1.1 Motion Model of the Container Transferring Mobile Robot
Let and be the linear velocities of the left and right driving wheels,
respectively. The track width is defined as 2L, and the half-track is L. Based on
rigid body kinematics, the linear velocity and angular velocity in the robot
body coordinate system can be derived from the wheel velocities as:
(1.1)
(1.2)
Then, the kinematic model in the global coordinate system can be written as
(1.3)
1.2 Lagrangian dynamics modeling
Differential drive mobile robots belong to non-holonomic constraint systems. Based
on the assumption that the 'lateral velocity of the vehicle body is zero', its non-
holonomic constraint can be expressed as:
(1.4)
Since the generalized coordinates of the robot are defined as , the
constraint can be written in Pfaffian form.
(1.5)
The dynamic equation with Lagrange multipliers can be expressed as Equation (1.6).
(1.6)
Where is the inertia matrix, denotes the Coriolis and centrifugal
Xuehong Zhu
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terms, is the generalized gravitational force, is the Lagrange multiplier for
the constraint forces, is the input transformation matrix, and
denotes the left and right wheel torques.
For low-speed trajectory tracking on a horizontal plane, the potential energy is
independent of the generalized coordinates; therefore, the generalized gravitational
force is zero .
Furthermore, in low-speed trajectory tracking scenarios, the Coriolis and centrifugal
terms, along with unmodeled dynamics such as rolling resistance and friction
variations, are relatively small compared to the dominant terms and can be lumped
into equivalent disturbances. Under these approximations, Equation (1.6) can be
simplified to:
(1.7)
To eliminate the constraint force terms from the dynamic equations, a null-space basis
matrix satisfying
=0 is introduced Equation (1.8).
(1.8)
Vector .
Pre-multiplying both sides of Equation (1.7) byyields:
(1.9)
By utilizing ,the constraint force terms can
be eliminated, yielding:
(1.10)
From Equation (1.8), we obtain:
(1.11)
Substituting into Equation (1.10) yields:
(1.12)
Where represents the velocity product terms associated with .
Considering the low-speed conditions and a small sampling period, these terms can
be approximated and lumped into the equivalent disturbances, yielding Equation
(1.13).
(1.13)
Defining the inertia matrix as
where m is the mass of the
vehicle body and I is the moment of inertia. By mapping the driving torques to the
longitudinal driving force and the yaw moment, the equivalent input mapping can be
expressed as:
Where r is the wheel radius, and L is the half-track width. Substituting these into
Equation (1.13) yields the dynamic equation of the differential-drive mobile robot as
Trajectory Tracking Control for a Container Transferring Mobile Robot Based on an Improved
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Equation (1.14)
(1.14)
can be equivalently expressed as:
(1.15)
Alternatively, the required driving torques for the mobile robot can be derived from
Equation (1.15), yielding:
(1.16)
1.3 Discretization of the Dynamic Model
For digital implementation, the continuous-time model in Eq. (1.15) is discretized
with a sampling period .
At discrete time instant (corresponding to continuous time ), let the system
state vector and the control input vector be defined respectively as:
,.
Where () denotes the posture of the robot in the world coordinate frame.
and are the linear and angular velocities of the vehicle body, respectively, and
represent the output driving torques of the right and left motors."
Given the high sampling frequency of the system, it is assumed that the rate of change
of the state remains approximately constant within a single sampling interval.
Consequently, this paper employs the first-order Forward Euler method to
approximately discretize the continuous-time dynamic model. Based on the finite-
difference definition of the derivative, the time derivative of the state variables can
be expressed as Equation (1.17)
(1.17)
Substituting the continuous-time nonlinear state equation into Equation
(1.17) and rearranging the terms yields the discrete-time recursive state model:
(1.18)
By substituting the kinematic model in Equation (1.3) and the dynamic model in
Equation (1.15) into Equation (1.18) and expanding the formulation, the system of
discrete-time nonlinear state transition equations for the differential-drive mobile
robot is obtained as follows:
Xuehong Zhu
8
(1.19)
2 Controller Design
2.1 Nonlinear Model Predictive Controller with Hard Constraints
The discrete model in Eq. (1.19) is used for prediction.
(2.1)
where denotes the predicted state at step made at time based on the
current measurement , and denotes the corresponding predicted
control input. The initial-state constraint is given by
(2.2)
Let denote the prediction horizon,
the reference trajectory, and
the reference input sequence. The standard NMPC employs a quadratic
cost function with fixed weighting, denoted by , to minimize the state-tracking
error and the control-input deviation:
(2.3)
Here
represents the weighted Euclidean norm. denotes the state
weighting matrix, which is used to regulate trajectory tracking accuracy.denotes
the terminal weighting matrix used to penalize the terminal tracking error. Under the
standard terminal conditions stated in Section 2.3, it also contributes to the
conditional closed-loop stability analysis. is the feedforward torque obtained
from inverse dynamics. The term
penalizes the deviation of the actual
torque from the reference torque, where is the control weighting matrix.
First, the physical constraints of the actuators must be considered. Limited by
the rated power of the brushless DC motors and their drivers, the output torque of the
right and left driving wheels, ,must be bounded within the saturation
limit to prevent overcurrent damage to the hardware. The mathematical
expression is given in Eq. (2.4):
(2.4)
Trajectory Tracking Control for a Container Transferring Mobile Robot Based on an Improved
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In addition to actuator constraints, geometric constraints are imposed to define the
admissible operating corridor of the robot under complex operating conditions. To
this end, this paper proposes a dynamic safety-envelope construction method based
on the reference trajectory: in the global trajectory-tracking task, a virtual safety
corridor with a total width of is dynamically generated along the normal
direction of the predefined reference trajectory. By combining the physical width of
the robot body, , with the unilateral safety margin reserved to
account for unmodeled errors, the maximum allowable lateral deviation of the
geometric center of the robot from the reference trajectory, namely the half-width of
the safety corridor, , can be strictly defined as:
=
(2.5)
The resulting virtual corridor is illustrated in Fig. 2.1.
Fig. 2.1. Schematic illustration of the virtual boundary
To prevent collisions between the robot body and the shelves, the lateral tracking
error of the robot, , is strictly constrained within the half-width of the safety
corridor, . By approximating the reference heading angle as the tangential angle
of the reference trajectory, we have:
atan2
(2.6)
The lateral error is defined as:
(2.7)
and, on this basis, the following hard constraint is imposed:
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In summary, the hard-constrained NMPC trajectory-tracking problem at time step
can be formulated as the following nonlinear programming problem:
hard
s.t.measured
(2.9)
By solving this optimization problem, the controller obtains the optimal control
sequence
Subsequently, only the first control input is
applied to the system, thereby implementing receding-horizon optimal control.
2.2 Nonlinear Model Predictive Controller with Soft Constraints
During narrow-corridor trajectory tracking, the robot may temporarily lie outside the
virtual corridor because of initial localization errors, path switching, or transient
disturbances. If Eq. (2.8) is retained as an unrelaxed hard constraint, the optimization
problem in Eq. (2.9) may have no feasible solution. To prevent the geometric corridor
constraint from directly causing optimization infeasibility, only the corridor
constraint is softened, whereas the actuator torque constraints remain hard. The
resulting control procedure is shown in Fig. 2.2.
Fig. 2.2. Flowchart of the controller
Trajectory Tracking Control for a Container Transferring Mobile Robot Based on an Improved
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A nonnegative slack-variable sequence is
introduced to soften the geometric hard constraint in Eq. (2.8):
(2.10)
Equation (2.10) allows the optimizer to maintain feasibility by selecting
when the original hard constraint cannot be temporarily satisfied. When ,
the predicted lateral error is allowed to exceed the original virtual-corridor boundary.
Therefore, the soft constraint improves optimization feasibility but does not strictly
guarantee that the robot always remains within . In this paper, the virtual
corridor is treated as a geometric constraint within the NMPC formulation rather than
as an independent strict safety guarantee.
(2.11)
In this paper, “adaptive” indicates that the penalty weight varies explicitly with the
instantaneous slack variable. It does not refer to an independent parameter-estimation
or learning law. When the slack variable increases, the corresponding penalty weight
increases automatically, thereby imposing a stronger penalty on large corridor
violations.
In Eq. (2.11), denotes the base quadratic penalty coefficient, and denotes
the adaptive barrier factor.
Combined with an -norm linear regularization term, the improved total penalty
cost function for the soft constraints, , is defined as:
soft
lin
(2.12)
By substituting Eq. (2.11) into the above expression and expanding it, the
mathematical essence of this adaptive mechanism can be clearly revealed:
soft
linbase
ba
(2.13)
Equation (2.13) shows that the slack-variable penalty consists of a linear term, a fixed
quadratic term, and a cubic term induced by the adaptive weight. For the conventional
fixed quadratic penalty, the gradient with respect to is . This
gradient approaches zero as , and therefore provides relatively limited
suppression of small nonzero slack values. In contrast, the gradient of the proposed
penalty function is given by
The linear term maintains a nonzero marginal penalty near , thereby helping
to suppress small but persistent slack values. As increases, the higher-order
term associated with causes the penalty gradient to increase nonlinearly,
strengthening the suppression of large constraint relaxations.
To ensure a fair comparison with the conventional fixed quadratic soft-constraint
formulation, all soft-constrained controllers employ the same prediction model, state
and input weighting matrices, prediction horizon, actuator constraints, base quadratic
Xuehong Zhu
12
penalty coefficient, and solver settings. Only the penalty components already defined
in Eq. (2.13) are enabled or disabled. When and , Eq. (2.13) reduces
to the conventional fixed quadratic penalty, corresponding to soft_fixed. When
and , the linear penalty is retained while the adaptive weighting is disabled,
corresponding to soft_no_adaptive. When and , both the linear
penalty and the adaptive weighting are enabled, corresponding to the complete
improved_NMPC. Therefore, the comparison between soft_fixed and
soft_no_adaptive isolates the contribution of the linear penalty term, whereas the
comparison between soft_no_adaptive and improved_NMPC isolates the
contribution of the adaptive weighting mechanism.
Taking the above soft-constraint mechanism into account, the overall cost function of
the improved NMPC, denoted by , is defined as:
imp
soft
(2.14)
The corresponding finite-horizon optimal control problem is formulated as follows:
imp
s.t.measured
(2.15)
By solving the above nonlinear programming problem, the controller obtains the
optimal control sequence
. The first element,
, is then
applied to the low-level drive system, and the same procedure is repeated at each
sampling instant to implement receding-horizon optimal control.
2.3 Stability Analysis
With the introduction of the soft-constraint mechanism in Eq. (2.10), the controller
relaxes the original geometric safety constraints by means of the slack variable ,
which substantially enhances the feasibility of the optimization problem under
extreme operating conditions. Accordingly, this section theoretically investigates the
improved soft-constrained NMPC from two aspects: recursive feasibility and closed-
loop behavior.
It is worth noting that the stability results presented below are derived under the
standard terminal-condition assumptions in NMPC, i.e., the existence of a terminal
local control law and a terminal invariant set, under which the optimal value function
can be employed as a candidate Lyapunov function[21]. Hence, the stability analysis
in this section should be regarded as a conditional theoretical justification. By contrast,
the recursive feasibility result follows directly from the intrinsic structure of the soft-
constrained problem formulated in this paper.
Trajectory Tracking Control for a Container Transferring Mobile Robot Based on an Improved
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In the hard-constrained NMPC formulation presented in Section 2.1, an excessively
large initial state error of the robot may lead to the absence of any admissible control
sequence within the prediction horizon that satisfies .
Consequently, the feasible set becomes empty, i.e., , and the
optimization solver may become stalled due to infeasibility. To address this issue, a
nonnegative slack variable is introduced in this paper, and the geometric
constraint is reformulated into the soft-constraint form shown in Eq. (2.10).
Theorem 2.1: Under the condition that only the input saturation constraints, the
system dynamic equality constraints, and the softened geometric constraints are taken
into account, with no additional hard state constraints or hard terminal constraints
imposed, the improved soft-constrained optimization problem defined in Eq. (2.15)
is always feasible for any initial state .
Proof:
For any given initial state , as long as the control inputs satisfy the actuator
physical saturation constraints, the corresponding predicted state sequence
can be uniquely generated from the discrete-time dynamic equations.
Accordingly, the values of the lateral error are determined. Since the slack
variable is only subject to the lower bound and has no upper bound, it is always
possible to choose
(2.16)
such that the soft constraint inequality is satisfied at all prediction steps. Therefore,
for any given initial state , one can always construct a feasible decision sequence
that satisfies all the constraints. Hence, the feasible set of the optimization
problem (2.15) is always nonempty, which implies that the problem remains feasible
for any initial state and therefore possesses recursive feasibility. Q.E.D.
From the foregoing proof, it can be seen that the introduction of slack variables
expands the originally finite feasible region truncated by hard constraints into a soft
feasible region over the entire state space, thereby preventing the softened corridor
constraint itself from causing infeasibility under the conditions of Theorem 2.1.
Numerical solver failure may still occur because of iteration limits, numerical
conditioning, inappropriate initialization, or implementation errors.
Under the assumptions of Theorem 2.1, the softened corridor constraint does not
make the optimization problem infeasible. However, feasibility does not
automatically imply closed-loop stability. To address this, the standard optimal value
function analysis framework commonly used in NMPC is introduced[22].
Assumption 2.1: There exist a terminal local control law and a terminal
invariant set such that: (1) for all , the input constraints are satisfied under
; (2) is positively invariant for the closed-loop system; (3) is contained
in the original hard-constrained feasible region, so that the terminal slack variable can
be selected as zero; and (4) the terminal weighting matrix satisfies the following
local descent condition. For the conditional analysis below, it is additionally assumed
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14
that the predicted terminal state satisfies
.
(2.17)
Let the optimal control sequence and the optimal slack sequence obtained at time be
denoted by and , respectively, and let the corresponding optimal cost be denoted
by
.
Theorem 2.2: Suppose that Assumption 2.1 holds. Then, the optimal value function
is non-increasing along the closed-loop trajectory. Furthermore, when the
original hard-constrained problem is feasible in the current neighborhood and the
exact penalty condition is satisfied, the closed-loop error system converges
asymptotically, and the optimal slack variables converge to zero.
Proof: After applying the optimal control input
at time , the system state
evolves to . At time , a shifted sequence is constructed as a suboptimal
candidate solution according to (2.18) and (2.19).
(2.18)
(2.19)
Note: Since the terminal state has entered the terminal invariant set, the slack variable
is set to zero.
Substituting this suboptimal sequence into the cost function at time , and
comparing it with the optimal cost at time , it follows from the terminal decrease
condition in Assumption 2.1 that:
(2.20)
Since the true optimal cost satisfies
, the Lyapunov
decrease condition is obtained as follows
(2.21)
Thus, the optimal value function is non-increasing. Under Assumption 2.1,
boundedness of the closed-loop trajectory, and the stated exact-penalty condition, the
tracking error and control-input deviation converge to zero, and the optimal slack
converges to zero once the original hard-constrained region becomes locally feasible.
Because no explicit terminal set is implemented in the numerical or experimental
controller, this result is a conditional stability property rather than an unconditional
guarantee.
3 Simulation Verification
The simulations comprised nominal feasible tracking tests, initial corridor-violation
tests, and multi-initial-condition recovery tests. The nominal tests compared the
standard and hard-constrained NMPC controllers, whereas the recovery tests
Trajectory Tracking Control for a Container Transferring Mobile Robot Based on an Improved
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compared the standard NMPC, soft_fixed, soft_no_adaptive, and improved_NMPC.
All simulations were performed in MATLAB R2022b using CasADi and IPOPT with
the MUMPS linear solver on a Windows 10 computer equipped with an Intel Core i5-
9300H CPU and 16 GB of RAM. The sampling period and prediction horizon were
s and , respectively. The IPOPT iteration limit and convergence
tolerance were set to 500 and , with warm-start initialization used for all
controllers.
For the straight-line tests, the nominal feasible scenario used an initial lateral
deviation of 0.04 m and a longitudinal velocity of 1.5 m/s, whereas the boundary-
recovery scenario used 0.10 m and 0.5 m/s, respectively. Comparisons were
performed only under identical conditions within each scenario. The circular
reference velocity was determined separately from its radius and angular velocity.
The parameters of the robot and the experimental setup are given in Table 3.1.
Table 3.1 Robot Parameters
Parameter
Symbol
Value
Unit
Vehicle width
1
m
Half track width
L
0.4
m
Wheel radius
r
0.1
m
Vehicle mass
m
200
kg
Moment of inertia
I
59.33
Virtual corridor width
1.2
m
Single-side safety
threshold
0.05
m
Based on Eq. (2.5) and the parameters in Table 3.1, a one-sided safety margin of 0.05
m yields an effective corridor half-width of
For the nominal feasible straight-line test, the initial lateral deviation was set to 0.04
m, which lies within the effective corridor. The standard and hard-constrained NMPC
controllers were compared in terms of tracking accuracy and computational efficiency.
The sampling period of the model predictive controller is set to , and the
prediction horizon is chosen as . The weighting matrices in the objective
function are selected as follows:
(3.1)
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16
To comprehensively assess the tracking performance and robustness of the controller
under different curvature conditions, two representative reference trajectories are
considered in this study: a straight line trajectory, a circular trajectory.
In trajectory tracking control of nonholonomic mobile robots, the reference trajectory
must satisfy the kinematic feasibility of the system. To this end, a virtual reference
vehicle model subject to the same nonholonomic constraints is introduced to generate
the desired states, and its state differential equations are defined as follows:
(3.2)
It follows from the geometric relationships that, for any given planar explicitly time-
parameterized path , the reference heading angle a nd the
reference control inputT can be obtained in a unified
manner by differentiation as follows:
arctan2
(3.3)
Performance metrics
To quantitatively evaluate the controller performance, let denote the total number
of simulation samples. The eight performance metrics used in the following
comparisons are formally defined as follows:
Here, denotes the empirical 95th percentile, is the indicator function
that equals 1 when its condition is satisfied and 0 otherwise, and denotes the
optimization time at the -th sampling instant in seconds. The lateral and
longitudinal errors are expressed in meters, the heading-angle errors in radians, the
boundary-violation rate in percent, and the average solution time in milliseconds.
(1) Straight-line trajectory
Based on the above framework, the straight-line reference trajectory with inclination
angle is defined in this paper as follows:
Trajectory Tracking Control for a Container Transferring Mobile Robot Based on an Improved
17
(3.4)
where denotes the constant reference linear velocity specified previously, and is
the inclination angle of the straight-line trajectory with respect to the positive -axis.
In this paper, is chosen, corresponding to a straight line at . Taking the
time derivative yields the components of the reference velocity as follows:
(3.5)
As shown in Figs. 3.1 and 3.2, in the nominal straight-line tracking scenario, the
initial lateral deviation is set to 0.04 m, and the initial and reference longitudinal
velocities are both set to 1.5 m/s, the simulated trajectories of the standard NMPC
and hard-constrained NMPC nearly overlap. This result indicates that the hard
geometric constraint does not noticeably affect tracking accuracy when the initial
state remains inside the admissible corridor. Under this nominal feasible condition,
both controllers achieve comparable tracking performance, while the hard-
constrained NMPC maintains the lateral deviation within the prescribed corridor
boundary.
Fig.3.1Straight-Line Trajectory
Tracking
Fig. 3.2 Straight-Line Tracking Error
As shown by the data in Table 3.2, the 95th-percentile error (P95) and the maximum
absolute error (Max) of the two controllers in the lateral, longitudinal, and heading
dimensions are exactly identical, and the boundary violation rate (viol%) is zero for
both.
The average solution times of the standard NMPC and the hard-constrained NMPC
are 6.93 ms and 7.50 ms, respectively. Although the hard-constrained formulation
introduces a slight increase in computational time, both values remain well below the
sampling period of 50 ms, indicating that the additional corridor constraint does not
compromise real-time feasibility under the nominal condition.
Table 3.1 Comparison of Comprehensive Performance Metrics for Straight-Line
Xuehong Zhu
18
Trajectory Tracking
Controller
latP95
latMax
lonP95
lonMax
psiP95
psiMax
viol%
t_ms
improved_
NMPC_hard
0.012
0.04
0.004
0.006
0.006
0.018
0
7.5
NMPC
0.012
0.04
0.004
0.006
0.006
0.018
0
6.93
(2) Circular trajectory
The circular reference trajectory can be derived from (3.3) as follows:
(3.6)
The velocity settings of 1.5 m/s and 0.5 m/s used in the preceding straight-line
experiments do not apply to the circular trajectory. For the circular reference
trajectory, the reference linear velocity is determined independently by .
With and the corresponding reference linear
velocity is 0.4 m/s.
Fig. 3.3 Circular Trajectory
Fig. 3.4 Circular Trajectory Error
It can likewise be observed from the figures and the table that, as long as no boundary
violation occurs, the responses of the two controllers remain highly consistent.
Therefore, no further detailed discussion is provided here.
Table 3.2 Comparison of Comprehensive Performance Metrics for Circular Trajectory
Tracking
Controller
latP95
latMax
lonP95
lonMax
psiP95
psiMax
viol%
t_ms
Trajectory Tracking Control for a Container Transferring Mobile Robot Based on an Improved
19
improved_
NMPC_hard
0.034
0.04
0.005
0.005
0.011
0.013
0
8.84
NMPC
0.034
0.04
0.005
0.005
0.011
0.013
0
8.07
Soft-constraint formulation:
To evaluate feasibility and boundary recovery under an initial corridor violation, a
low-speed straight-line scenario was considered with an initial lateral deviation of
0.10 m, exceeding the corridor half-width of 0.05 m. The initial and reference
longitudinal velocities were both set to 0.5 m/s. Consequently, the hard-constrained
NMPC was infeasible at the initial sampling instant and was excluded from the
closed-loop comparisons.
According to Eq. (2.13), the base quadratic, adaptive, and linear penalty coefficients
were set to 50,000, 500, and 200, respectively. The standard NMPC, soft_fixed,
soft_no_adaptive, and improved_NMPC were evaluated using identical models,
trajectories, controller settings, and actuator constraints. As shown in Figs. 3.5–3.8,
all soft-constrained controllers remained feasible, with improved_NMPC providing
the fastest lateral-error recovery.
Fig. 3.5 Soft-Constrained Straight-
Line Trajectory
Fig. 3.6 Lateral Error under Constrained
Straight-Line Tracking
As shown in Figs. 3.5–3.8, the standard NMPC produces the slowest lateral-error
reduction because it does not explicitly account for the virtual corridor boundary. The
fixed-weight quadratic soft NMPC and the non-adaptive linear-quadratic soft NMPC
reduce the lateral deviation more rapidly, while their responses remain highly similar.
The proposed adaptive soft NMPC produces the fastest boundary-recovery response
and returns the lateral deviation to the effective corridor in approximately 1 s.
However, this faster lateral correction is accompanied by larger transient longitudinal
and heading-angle errors. In particular, the proposed controller reaches a maximum
longitudinal error of 0.217 m and a maximum heading-angle error of 0.215 rad.
Therefore, the adaptive mechanism improves boundary-recovery performance at the
cost of increased transient motion adjustment, rather than improving all tracking-error
Xuehong Zhu
20
indices simultaneously
Fig. 3.7 Longitudinal Error under Soft-
Constrained Straight-Line Tracking
Fig.3.8 Heading-Angle Error under Soft-
Constrained Straight-Line Tracking
As in Table 3.4, the hard-constrained NMPC was infeasible at the initial sampling
instant. Compared with the standard NMPC, soft_fixed reduced latP95 from 0.086 to
0.025 m and the violation rate from 16.48% to 2.22%. Soft_no_adaptive provided a
modest further reduction to 0.024 m and 2.06%, respectively, whereas
improved_NMPC achieved the lowest values of 0.004 m and 1.58%. However,
improved_NMPC produced the largest lonMax and psiMax values of 0.217 m and
0.215 rad. Its solution time of 11.59 ms remained below the 50 ms sampling period.
Table 3.4 Comparison of Comprehensive Performance Metrics for Soft-
Constrained Straight-Line Trajectory Tracking
Controller
latP95
latMax
lonP95
lonMax
psiP95
psiMax
viol%
t_ms
Improved
_NMPC_hard
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
NMPC
0.086
0.10
0.010
0.012
0.023
0.027
16.48
6.77
improved_NMPC
0.004
0.1
0.007
0.217
0.006
0.215
1.58
11.59
soft_fixed
0.025
0.100
0.006
0.099
0.007
0.181
2.22
9.76
soft_no_adaptive
0.024
0.100
0.006
0.103
0.006
0.185
2.06
11.59
In the circular-trajectory experiment, the hard-constrained NMPC is again infeasible
at the initial sampling instant and is therefore excluded from the closed-loop curve
comparison. As shown in Figs. 3.9–3.12, the standard NMPC exhibits the slowest
lateral-error reduction. Both soft_fixed and soft_no_adaptive substantially accelerate
boundary recovery, whereas their lateral, longitudinal, and heading-angle responses
remain close to each other. The proposed improved_NMPC produces the fastest
lateral-error reduction and the lowest boundary-violation exposure, although it
generates larger transient longitudinal and heading-angle errors during the initial
recovery stage.
Trajectory Tracking Control for a Container Transferring Mobile Robot Based on an Improved
21
Fig. 3.9 Soft-Constrained Circular
Trajectory
Fig. 3.10 Lateral Error in Circular Trajectory
Tracking
It can also be observed from Figs. 3.11 and 3.12 that, in order to correct the initial
lateral deviation, the soft-constrained NMPC produces relatively large longitudinal
and heading-angle errors during roughly the first 1 s. The larger transient longitudinal
and heading-angle errors reflect the control effort required for rapid lateral recovery.
These transient responses remain bounded in the present numerical simulations, but
their practical acceptability should be further evaluated through platform experiments.
Fig. 3.11 Longitudinal Error under Soft-
Constrained Circular Tracking
Fig. 3.12 Heading-Angle Error under
Soft-Constrained Circular Tracking
As shown in Table 3.5, soft_fixed reduced latP95 and the violation rate from 0.087 m
and 15.85% for the standard NMPC to 0.025 m and 2.22%, respectively.
Improved_NMPC achieved the lowest values of 0.017 m and 1.58%, but produced
the largest transient longitudinal and heading errors.
Table 3.5 Comparison of Comprehensive Performance Metrics for Soft-
Constrained Circular Trajectory Tracking
Controller
latP95
latMax
lonP95
lonMax
psiP95
psiMax
viol%
t_ms
improved_
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
Xuehong Zhu
22
NMPC_hard
NMPC
0.087
0.100
0.016
0.016
0.021
0.024
15.85
8.63
improved_NMPC
0.017
0.100
0.008
0.227
0.003
0.200
1.58
11.70
soft_fixed
0.025
0.100
0.008
0.129
0.010
0.165
2.22
10.96
soft_no_adaptive
0.022
0.100
0.008
0.133
0.009
0.170
2.22
11.85
Multi-Initial-Condition Boundary-Recovery Tests
The preceding boundary-recovery tests used fixed initial lateral and heading errors.
To examine whether the observed recovery performance depends on a single initial
state, additional simulations were conducted under multiple initial lateral and heading
errors.
Ten initial conditions were considered for each of the straight-line and circular
trajectories. Let denote the initial lateral error and initial heading error,
respectively. The selected initial conditions were:
Here, and are expressed in meters and radians, respectively. The selected
conditions cover both sides of the reference trajectory and positive and negative initial
heading errors. Because m in all tests, the hard-constrained
NMPC was initially infeasible and is therefore reported as N/A.
The standard NMPC, soft_fixed, soft_no_adaptive, and improved_NMPC were
evaluated using identical models, controller settings, and solver parameters, with only
the penalty components defined in Section 2.2 varied. The straight-line tests used
initial and reference longitudinal velocities of 0.5 m/s, whereas the circular reference
velocity was determined by the prescribed radius and angular velocity. Table 3.6
reports the mean and standard deviation of the eight performance metrics over ten
initial conditions.
Table 3.6 Statistical Results of Comprehensive Performance Metrics under
Multi-Initial-Condition Boundary-Recovery Tests
(a) Straight-line trajectory
Controller
latP95
latMax
lonP95
lonMax
psiP95
psiMax
viol%
t_ms
improved_
NMPC_hard
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
NMPC
0.068
±0.019
0.084
±0.019
0.009
±0.002
0.010 ±
0.003
0.019
±0.005
0.044±
0.012
11.06
±6.03
6.75
±0.12
soft_fixed
0.032
±0.007
0.081
±0.019
0.006
±0.001
0.055 ±
0.044
0.009
±0.002
0.120 ±
0.059
2.11
±0.37
9.82
±0.23
Trajectory Tracking Control for a Container Transferring Mobile Robot Based on an Improved
23
(b) Circular trajectory
Controller
latP95
latMax
lonP95
lonMax
psiP95
psiMax
viol%
t_ms
improved_
NMPC_hard
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
NMPC
0.071
±0.019
0.083
±0.019
0.013
±0.004
0.014
±0.004
0.014±
0.006
0.043
±0.014
12.58
±6.87
8.18
±
0.14
soft_fixed
0.037
±0.007
0.081
±0.019
0.009
±0.002
0.064
±0.046
0.008±
0.004
0.120
±0.065
2.31
±0.34
10.47
±0.33
soft_no_adaptive
0.035
±0.008
0.081
±0.019
0.008
±0.002
0.068
±0.047
0.008±
0.003
0.125
±0.067
2.19
±0.31
11.49
±0.09
improved_NMPC
0.032
±0.011
0.081
±0.019
0.011
±0.009
0.163
±0.141
0.007±
0.003
0.132
±0.073
1.66
±0.23
11.47
±0.20
The hard-constrained NMPC was infeasible for all tested initial conditions. For the
straight-line trajectory, soft_fixed reduced the mean latP95 and violation rate from
0.068 ± 0.019 m and 11.06 ± 6.03% for the standard NMPC to 0.032 ± 0.007 m and
2.11 ± 0.37%, respectively. Soft_no_adaptive provided a modest further improvement,
whereas improved_NMPC achieved the lowest values of 0.023 ± 0.009 m and 1.52 ±
0.17%.
For the circular trajectory, the mean latP95 and violation rate decreased from 0.071 ±
0.019 m and 12.58 ± 6.87% for the standard NMPC to 0.037 ± 0.007 m and 2.31 ±
0.34% for soft_fixed. Improved_NMPC further reduced these values to 0.032 ± 0.011
m and 1.66 ± 0.23%, confirming the additional, although modest, benefit of the linear
and adaptive penalty terms.
Because latMax was primarily determined by the prescribed initial error, it was not
used to assess recovery performance. Improved_NMPC achieved better lateral
recovery at the cost of larger transient longitudinal errors, with mean lonMax values
of 0.115 ± 0.090 m and 0.163 ± 0.141 m for the straight-line and circular trajectories,
respectively. Its mean solution times remained below the 50 ms sampling period for
both trajectories.
4 Hardware Experimental Validation
4.1 Experimental Platform and Parameter Settings
Hardware trajectory-tracking experiments were conducted on a differential-drive
soft_no_adaptive
0.030
±0.007
0.081
±0.019
0.006
±0.001
0.058 ±
0.046
0.008
±0.002
0.125 ±
0.060
2.04
±0.34
11.43
±0.20
improved_NMPC
0.023
±0.009
0.081
±0.019
0.006
±0.002
0.115 ±
0.090
0.008
±0.005
0.157 ±
0.085
1.52
±0.17
11.41
±0.22
Xuehong Zhu
24
container transferring mobile robot to evaluate the practical implementability of the
proposed controller. The onboard control system consisted of an Advantech ARK-
1250 industrial computer with an Intel Core i5-1145G7E processor and 8 GB of RAM,
running a ROS-based control algorithm. The computer transmitted wheel-torque
commands via a CAN bus to two Kinco FD134S-CB-000 servo drives, which
controlled Kinco SMC80S-0075-30MAK-5DSU motors through 1:18 reduction
mechanisms.
The hardware experiments used the same robot geometry and virtual-corridor
parameters as the simulations (Table 3.1). The sampling period was 0.05 s, the
prediction horizon was 20, and the wheel-torque limits were ±5 N·m. Each
experiment lasted approximately 64 s at a reference linear velocity of 0.08 m/s, with
an initial lateral error of −0.08 m and an initial heading error of 0 rad.
As shown in Fig. 4.1, the platform comprises a differential-drive chassis and an upper
container-handling mechanism. Only the chassis was operated because the
experiments focused on trajectory-tracking performance. The CAN topology is
shown in Fig. 4.2; the left and right drives, assigned node IDs 1 and 3, respectively,
performed closed-loop motor control using encoder feedback.
Fig. 4.1 Experimental Platform of the
Container Transferring Mobile Robot
Fig. 4.2 CAN-Based Control and Drive
System Architecture
4.2 Experimental Conditions and Controller Settings
Straight-line and circular trajectory-tracking experiments were conducted to evaluate
controller feasibility and boundary recovery from an initial state outside the
prescribed safety threshold. The standard NMPC, soft_fixed, and improved_NMPC
were compared under identical experimental conditions. The hard-constrained NMPC
was excluded because the initial lateral error exceeded the corridor threshold,
rendering its optimization problem infeasible at the initial sampling instant.
Both trajectories were tracked at a reference linear velocity of 0.08 m/s for
Trajectory Tracking Control for a Container Transferring Mobile Robot Based on an Improved
25
approximately 64 s, with an initial lateral error of −0.08 m and a safety threshold of
0.05 m. The circular trajectory had a radius of 2 m and a reference angular velocity
of approximately 0.04 rad/s. Because of space and safety limitations, the robot
traversed an arc of approximately 147° rather than a complete circle. The remaining
experimental parameters are summarized in Table 4.1.
The straight-line and circular reference trajectories were generated using Eqs. (3.4)
and (3.6), respectively, with for the straight-line trajectory.
Table 4.1 Hardware Experimental Parameters
Parameter
Straight line
Circular trajectory
Sampling time/s
0.05
0.05
Prediction horizon
20
20
Reference linear velocity/(m/s)
0.08
0.08
Reference angular velocity/(rad/s)
0
0.04
Trajectory radius/m
—
2.0
Initial lateral error/m
−0.08
−0.08
Initial heading error/rad
0
0
Lateral safety threshold/m
0.05
0.05
Wheel-torque limit/(N·m)
±5
±5
Experimental duration/s
Approximately 64
Approximately 64
Considering the actuator response, ground resistance, and communication delay
of the physical robot, the controller weighting matrices were adjusted from thei r
simulation values. The three control strategies employed:
(4.1)
The soft-constraint parameters were set to for
soft_fixed and to for improved_NMPC,
according to Eq. (2.13). Because the controller weights were tuned separately on the
hardware platform, these experiments assess practical implementation and overall
performance rather than provide a controlled ablation. The contributions of the
individual penalty components are evaluated using identical weighting matrices in
the simulations in Section 3. Hardware performance was assessed using the eight
metrics defined in Section 3.
4.3 Straight-Line Trajectory Tracking Experiment
Figure 4.3 compares the trajectories and tracking errors of the three controllers in the
straight-line experiment. All controllers maintained closed-loop operation without
solver failure or fallback activation. The standard NMPC gradually reduced the lateral
error but did not remain within the safety threshold by the end of the experiment,
Xuehong Zhu
26
whereas both soft-constrained controllers recovered within the threshold, with
improved_NMPC showing slightly faster recovery. Their initially larger longitudinal
and heading errors subsequently decreased and fluctuated around zero.
(a) Straight-line trajectory
(b) Lateral error
(c)Longitudinal error
(d)Heading-angle error
Fig. 4.3 Hardware Experimental Results of Straight-Line Trajectory Tracking:
(a) trajectories; (b) lateral errors; (c) longitudinal errors; (d) heading-angle errors.
To further quantitatively compare the hardware performance of the three controllers,
Table 4.2 lists the eight performance metrics defined in Section 3.
Table 4.2 Comparison of Comprehensive Performance Metrics for Hardware
Straight-Line Trajectory Tracking
Controller
latP95
latMax
lonP95
lonMax
psiP95
psiMax
viol%
t_ms
NMPC
0.080
0.081
0.022
0.041
0.017
0.031
100.00
7.80
soft_fixed
0.079
0.080
0.023
0.040
0.115
0.157
10.77
9.80
improved_NMPC
0.077
0.080
0.039
0.061
0.103
0.161
9.13
13.13
Compared with soft_fixed, improved_NMPC reduced the violation rate from 10.77%
to 9.13% and latP95 from 0.079 to 0.077 m, but increased lonP95 from 0.023 to 0.039
m and lonMax from 0.040 to 0.061 m. All controllers completed the computation
Trajectory Tracking Control for a Container Transferring Mobile Robot Based on an Improved
27
within the 50 ms sampling period.
4.4 Circular Trajectory Tracking Experiment
Figure 4.4 compares the trajectories and tracking errors of the three controllers in the
circular experiment. The standard NMPC exhibited a persistent lateral deviation and
did not enter the prescribed safety threshold. Both soft-constrained controllers
recovered toward the reference trajectory, with improved_NMPC showing slightly
faster lateral-error recovery. Their transient longitudinal and heading errors gradually
decreased after the initial correction stage.
(a) Circular trajectory
(b)Lateral error
(c)Longitudinal error
(d)Heading-angle error
Fig. 4.4 Hardware Experimental Results for Circular Trajectory Tracking: (a)
trajectories; (b) lateral errors; (c) longitudinal errors; and (d) heading-angle errors.
Table 4.3 Comparison of Comprehensive Performance Metrics for Hardware
Circular Trajectory Tracking
Controller
latP95
latMax
lonP95
lonMax
psiP95
psiMax
viol%
t_ms
NMPC
0.123
0.124
0.053
0.059
0.059
0.095
100.00
7.45
soft_fixed
0.079
0.080
0.088
0.112
0.027
0.075
11.32
9.95
Xuehong Zhu
28
improved_NMPC
0.075
0.080
0.077
0.099
0.049
0.079
9.29
13.25
Compared with soft_fixed, improved_NMPC reduced latP95 from 0.079 to 0.075 m,
the violation rate from 11.32% to 9.29%, and the recovery time from 7.25 to 5.95 s.
It also reduced longitudinal errors but produced slightly larger heading errors. All
controllers completed the computation within the 50 ms sampling period without
solver failure or fallback activation.
5 Conclusions and Future Work
An improved soft-constrained NMPC method was developed for trajectory tracking
of a container transferring mobile robot in narrow corridors. A virtual-corridor
constraint was constructed based on the robot width, aisle width, and safety margin.
Slack variables were introduced to maintain optimization feasibility, while linear and
slack-dependent penalties were used to strengthen boundary recovery. Simulations
showed that the proposed method improved trajectory-tracking accuracy and
boundary-recovery performance, reducing latP95 and the boundary-violation rate,
although faster recovery produced larger transient longitudinal and heading errors.
Hardware experiments confirmed recovery from an initial lateral error of −0.08 m.
Compared with soft_fixed, improved_NMPC slightly improved trajectory-tracking
accuracy and reduced the recovery time and boundary-violation rate for both straight-
line and circular trajectories. Its average solution time remained below the 50 ms
sampling period, demonstrating real-time implementability.
Nevertheless, the softened virtual corridor does not provide an absolute safety
guarantee. Future work will include repeated hardware tests under unified controller
weights and evaluations at different velocities, payloads, and external disturbances.
Fund project:Research Initiation Fund Project of Civil Aviation University of
China (No. 2020KYQD84)
Fundamental Research Funds for the Central Universities (No. 3122022052)
References
[1] Klančar, G., & Seder, M. (2022). Coordinated multi-robotic vehicles navigation
and control in shop floor automation. Sensors, 22(4), 1455.
[2] Achirei, S. D., Mocanu, R., Popovici, A. T., & Dosoftei, C. C. (2023). Model-
predictive control for omnidirectional mobile robots in logistic environments
based on object detection using CNNs. Sensors, 23(11), 4992.
[3] Lucet, Eric, Antoine Lucazeau, and Jason Chemin. "Autonomous Forklift
Navigation Inside a Cluttered Logistics Factory." ICINCO (2). 2024.
[4] Zhang, L., Ao, W., Zhang, H., Liu, P., Li, Z., & Minchala, L. I. (2024). Dynamics
modeling and trajectory tracking control of a life support robot using sliding-
mode control with extended state observer. Journal of the Franklin
Institute, 361(12), 107013.
[5] Moritz, J., Musa, M., & Wejinya, U. (2024). Design and modeling of a two-
wheeled differential drive robot. International Journal of Control, Automation
Trajectory Tracking Control for a Container Transferring Mobile Robot Based on an Improved
29
and Systems, 22(7), 2273-2282.
[6] Liu, Q., & Cong, Q. (2022). Kinematic and dynamic control model of wheeled
mobile robot under internet of things and neural network. The Journal of
Supercomputing, 78(6), 8678-8707.
[7] Köhler, J., Müller, M. A., & Allgöwer, F. (2024). Analysis and design of model
predictive control frameworks for dynamic operation—An overview. Annual
Reviews in Control, 57, 100929.
[8] Wang, J., Liu, Z., Chen, H., Zhang, Y., Zhang, D., & Peng, C. (2023). Trajectory
tracking control of a skid-steer mobile robot based on nonlinear model predictive
control with a hydraulic motor velocity mapping. Applied Sciences, 14(1), 122.
[9] Peng, J., Xiao, H., & Lai, G. (2024). Nonlinear disturbance observer incorporated
model predictive strategy for wheeled mobile robot’s trajectory tracking
control. International Journal of Control, Automation and Systems, 22(7), 2251-
2262.
[10] Tang, M., Zhang, Y., Yu, S., Li, J., & Tang, K. (2025). Trajectory tracking model
predictive control for mobile robot based on deep Koopman operator
modeling. Robotics and Autonomous Systems, 105152.
[11] Tang, Jiawei, et al. "GMPC: Geometric model predictive control for wheeled
mobile robot trajectory tracking." IEEE Robotics and Automation Letters 9.5
(2024): 4822-4829.
[12] Fnadi, M., Du, W., Plumet, F., & Benamar, F. (2021). Constrained model
predictive control for dynamic path tracking of a bi-steerable rover on slippery
grounds. Control Engineering Practice, 107, 104693.
[13] Nam, N. N., & Han, K. (2024). Path-tracking robust model predictive control of
an autonomous steering system using LMI optimization with independent
constraints enforcement. International Journal of Control, Automation and
Systems, 22(11), 3352-3363.
[14] Jian, Zhuozhu, et al. "Dynamic control barrier function-based model predictive
control to safety-critical obstacle-avoidance of mobile robot." 2023 IEEE
international conference on robotics and automation (ICRA). Ieee, 2023.
[15] Khan, S., Guivant, J., & Li, X. (2022). Design and experimental validation of a
robust model predictive control for the optimal trajectory tracking of a small-scale
autonomous bulldozer. Robotics and Autonomous Systems, 147, 103903.
[16] Wabersich, K. P., Krishnadas, R., & Zeilinger, M. N. (2021). A soft constrained
MPC formulation enabling learning from trajectories with constraint
violations. IEEE Control Systems Letters, 6, 980-985.
[17] Gracia, V., Krupa, P., Limon, D., & Alamo, T. (2024). Implementation of soft-
constrained MPC for tracking using its semi-banded problem structure. IEEE
Control Systems Letters, 8, 1499-1504.
[18] Liu, Huajian, et al. "Optimization-based local planner for a nonholonomic
autonomous mobile robot in semi-structured environments." Robotics and
Autonomous Systems 171 (2024): 104565.
[19] Sun, Y., Yang, J., Zhao, D., Okonkwo, M. C., Zhang, J., Wang, S., & Liu, Y.
(2024). Enhancing stability and safety: A novel multi‐constraint model predictive
control approach for forklift trajectory. IET Cyber‐Systems and Robotics, 6(4),
e70004.
Xuehong Zhu
30
[20] Magalhães, André Chaves, and Luciano Cunha de Araujo Pimenta. "Distributed
Model Predictive Control for Safety-Critical Obstacle Avoidance in Multi-Robot
Systems with Social Preferences." IFAC-PapersOnLine 59.21 (2025): 56-61.
[21] Mayne, D. Q., Rawlings, J. B., Rao, C. V., & Scokaert, P. O. (2000).
Constrained model predictive control: Stability and
optimality. Automatica, 36(6), 789-814.
[22] Chen, H., & Allgöwer, F. (1998). A quasi-infinite horizon nonlinear model
predictive control scheme with guaranteed stability. Automatica, 34(10), 1205-
1217.
[23] Yang, H., Guo, M., Xia, Y., & Cheng, L. (2018). Trajectory tracking for
wheeled mobile robots via model predictive control with softening
constraints. IET Control Theory & Applications, 12(2), 206-214.
[24] Zheng, W., & Zhu, B. (2021). Stochastic time-varying model predictive control
for trajectory tracking of a wheeled mobile robot. Frontiers in Energy
Research, 9, 767597.
