Kinematic modeling of differential-drive mobile robots

To achieve pose tracking of wheeled differential-drive mobile robots, a standard nonholonomic kinematic model is used to describe the robot planar pose in a world coordinate system. This model consists of three states and two velocity inputs derived from geometric kinematic relationships. Fig. 1 illustrates the motion decomposition, while Table 2 summarizes the key symbols used in the kinematic model.

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Table 2. Symbol definitions for the robot kinematic model.

https://doi.org/10.1371/journal.pone.0354699.t002

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Fig 1. Kinematic model of the robot in the world coordinate frame.

https://doi.org/10.1371/journal.pone.0354699.g001

Let the robot pose be represented by

(1)

where x and y are the coordinates of the robot center of mass in the world frame , and is the heading angle measured counterclockwise from the axis. The differential-drive platform satisfies the nonholonomic constraint that the lateral velocity in the body-fixed frame is zero.

In an ideal kinematic model, the velocity input is usually written as . In the proposed hierarchical controller, however, the velocity generated by the outer-loop MPC is not necessarily equal to the velocity actually applied to the robot because of event-triggered input holding, inner-loop velocity servoing, input saturation, and actuator dynamics. Therefore, the actual velocity applied to the robot is denoted by

(2)

Using a planar rotation by to transform the body-frame velocity into the inertial frame, the continuous-time kinematic model driven by the actual actuator velocity is

(3)

Eq (3) describes the continuous-time kinematic model of the differential-drive mobile robot in the world frame. Its compact matrix form is

(4)

Eqs (3)–(4) provide the basis for the subsequent prediction model and controller design. The key point is that the physical plant is driven by , whereas the MPC optimizes a nominal velocity setpoint. This distinction is retained in the following modeling process to explicitly represent actuator mismatch.

The modeling assumptions used in this study are summarized as follows. First, lateral wheel–ground slip is neglected, and the differential-drive robot is assumed to satisfy the nonholonomic rolling constraint. Second, the outer-loop MPC is designed based on the kinematic model and the local error-augmented linearized prediction model. Third, actuator delay, saturation, and unmodeled fast dynamics are represented by a bounded mismatch between the MPC velocity setpoint and the actual actuator velocity. This mismatch is further regulated by the inner-loop PID velocity servo introduced in the controller design section.

Input signal chain and actuator model

To clarify the hierarchical control architecture, three velocity signals are distinguished in this study.

First, the outer-loop MPC generates the velocity setpoint

(5)

This signal is the nominal control input optimized by the MPC. In the event-triggered scheme, is updated only when the triggering condition is satisfied or when the maximum holding time is reached. Otherwise, the previously generated setpoint is held.

Second, the actuator command after the inner-loop controller and saturation is denoted as

(6)

For the proposed ET-MPC-PID controller, is generated by the PID velocity servo and then limited by actuator saturation. For the ET-MPC ablation controller without the PID inner loop, the command input is obtained by directly saturating the MPC setpoint.

Third, the actual velocity applied to the robot is

(7)

This actual velocity drives the plant model in Eq (3). In general,

(8)

because the signal chain includes event-triggered setpoint holding, inner-loop servoing, actuator saturation, and first-order actuator dynamics.

The admissible MPC velocity-setpoint bounds are expressed as

(9)

where

(10)

The input-rate constraint used in the MPC is written as

(11)

where

(12)

The actuator command is subject to component-wise physical saturation:

(13)

where denotes the unsaturated output of the inner-loop velocity servo and

(14)

The saturation operator is defined as

(15)

with

(16)

To represent actuator lag and low-level response delay, the actuator dynamics are modeled as a first-order discrete-time system:

(17)

where

(18)

Here, and are the time constants of the linear and angular velocity channels, respectively. A larger time constant indicates a slower actuator response. This model explains why the actual velocity may lag behind the setpoint even when the command input satisfies the physical constraints.

The actuator-side execution mismatch is defined as

(19)

This quantity represents the discrepancy between the held MPC velocity setpoint and the velocity actually applied to the robot. It captures the effects of actuator lag, command saturation, and unmodeled fast execution dynamics for a given setpoint.

The effect of retaining an outdated outer-loop decision is distinguished separately. Let denote the velocity setpoint that would be obtained if the MPC optimization were solved at time k. The holding-induced decision discrepancy is defined as

(20)

Thus, characterizes the deviation caused by retaining the previously optimized setpoint, whereas characterizes the actuator-side deviation relative to that held setpoint. Their effects are introduced separately in the subsequent local perturbation analysis.

Model discretization and state-space formulation

To enable online implementation on a digital controller, the continuous-time kinematic model in Eq (3) is discretized with sampling period . The actual plant model is obtained by forward Euler discretization as

(21)

This equation describes the physical robot motion driven by . In contrast, the MPC prediction model is constructed with respect to the nominal velocity setpoint . This separation allows the controller to optimize a constrained nominal input while the plant model still reflects actuator lag and saturation.

For trajectory tracking, let

(22)

denote the reference pose and reference velocity at time k, respectively. The world-frame pose tracking error is defined as

(23)

where

(24)

and maps an angle to . This avoids artificial discontinuities when the heading angle crosses .

To explicitly incorporate input-rate constraints in the MPC formulation, the control increment is selected as the decision variable:

(25)

The previous setpoint is appended to the error state to form the augmented state vector

(26)

A first-order Taylor expansion of the nominal discrete-time kinematic model around the reference pair , neglecting higher-order terms, yields an error-augmented linear time-varying model:

(27)

In Eq (27), denotes the MPC prediction output, namely the tracking error expressed in a reference-aligned local frame. The matrix is constructed from the rotation matrix , which is defined in Eq (28).

The system matrix and input matrix are computed from the Jacobians evaluated at the reference pair :

To ensure consistency among prediction, cost evaluation, and event-triggering feedback, the MPC output, the cost-function error, and the triggering error are all defined using the geometric tracking error expressed in a reference-aligned local frame. The corresponding reference-aligned tracking error is

(28)

With this definition, the MPC output satisfies .

Because the plant is driven by while the prediction model uses , the actual model can be written as the nominal model plus an actuator-side execution mismatch. Substituting

(29)

into Eq (21) gives

(30)

where

(31)

Equation (30) shows that the difference between the nominal MPC prediction model and the actual plant can be represented by a bounded actuator-side input mismatch when the command input, actuator saturation, and actuator dynamics are bounded. The holding-induced decision discrepancy in Eq (20) is treated separately because it arises from intermittent outer-loop optimization.

Accordingly, in Eq (27) and the predicted error terms used in the cost function are defined in the same reference-aligned local coordinates, thereby maintaining internal consistency of MPC prediction, cost evaluation, and feedback. Because is discontinuous at , the linearization and optimization are intended to operate in a small-error region. When the heading error approaches the boundary of this region, an event-triggered update and re-linearization are used to restore the validity of the local prediction model.

The trajectory tracking control problem: Formulation and analysis

This study addresses trajectory tracking of a differential-drive mobile robot under input-magnitude constraints, input-rate constraints, actuator saturation, and first-order actuator dynamics. Consistent with the previous subsection, the tracking error is evaluated in the reference-aligned local frame using defined in Eq (28).

At time k, the MPC predicts the future behavior over a finite prediction horizon of length N. The notation denotes the i-step-ahead prediction made at time k, where . In particular, denotes the predicted MPC output and denotes the predicted reference-aligned tracking error at step k + i, both computed based on information available at time k. In the prediction recursion, the propagated state corresponds to the LTV augmented world-frame error state generated by Eq (27), and the corresponding reference-aligned tracking error is obtained through the time-varying coordinate transformation along the reference trajectory:

(32)

where is the reference heading at step k + i. Because the transformation varies along the prediction horizon, the i-step prediction uses rather than a fixed .

For compactness, the admissible set of the MPC velocity setpoint and the admissible set of the input increment are expressed as

(33)

where and denote the MPC setpoint and setpoint-increment constraint sets, respectively.

The trajectory-tracking control problem considered in this study can now be stated as follows. Given the reference trajectory and the reference velocity , design a hierarchical controller that generates the outer-loop velocity setpoint , the actuator command , and the actual applied velocity , such that the following objectives are achieved:

1. The reference-aligned tracking error remains small and ultimately bounded.
2. The MPC velocity setpoint and input increment satisfy the constraints in Eq (33).
3. The actuator command satisfies the physical saturation limits.
4. The number of outer-loop MPC optimizations is reduced by updating the MPC only when the event-triggering condition is satisfied or when the maximum holding time is reached.
5. The holding-induced decision discrepancy and the actuator-side execution mismatch are explicitly distinguished. The former is regulated by the event-update policy, whereas the latter is reduced by the inner-loop velocity servo and evaluated in the stability and simulation analyses.

Periodic MPC typically achieves high tracking accuracy but requires solving an optimization problem at every sampling instant. By contrast, the event-triggered hierarchical MPC-PID framework developed in the next section introduces a tunable trade-off between computational efficiency and tracking performance. The modeling formulation above also clarifies that the proposed method is not designed for an ideal plant where the MPC input is applied instantaneously; instead, it addresses the more practical case where the robot is driven by delayed and saturated actuator velocities.

Controller design

Building on the error-augmented discrete-time model developed in Section 2, this section presents the proposed event-triggered MPC–PID trajectory-tracking controller. The objective is to reduce unnecessary online MPC optimization while maintaining practically acceptable tracking performance in the presence of input constraints, actuator saturation, setpoint holding, and finite actuator response speed. The proposed contribution does not arise from the cascaded MPC–PID architecture alone. Instead, it lies in explicitly coupling an event-triggered constrained MPC outer loop with a fixed-period PID velocity servo and in characterizing the resulting execution mismatch caused by intermittent optimization, actuator lag, and saturation.

The overall architecture of the proposed ET–MPC–PID framework is shown in Fig 2. The outer-loop event-triggering mechanism determines whether the MPC optimization should be solved at the current sampling instant. When the triggering condition is not satisfied, the previously optimized velocity setpoint is retained by the memory unit. The inner-loop PID velocity servo operates at every sampling instant and regulates the actuator response toward the held MPC setpoint. Its raw output is first limited by the saturation block and subsequently filtered by the first-order actuator dynamics. Therefore, the velocity applied to the robot kinematic model is , rather than the nominal MPC setpoint .

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Fig 2. Overall architecture of the proposed ET–MPC–PID trajectory-tracking framework.

https://doi.org/10.1371/journal.pone.0354699.g002

All modules are implemented with the same sampling period . The primary feedback signal is the reference-aligned geometric tracking error defined in Eq. (28). For the MPC outer loop, both the prediction output and the tracking-error term in the cost function are expressed in the reference-aligned local frame through the output mapping in Eq. (27). The velocity and input-rate constraints are consistent with the modeling definitions in Eqs. (9)–(11). The corresponding numerical parameters are listed in Table 3.

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Table 3. Simulation, controller, and computational settings. Values in parentheses refer to the Eight scenario.

https://doi.org/10.1371/journal.pone.0354699.t003

Outer-loop MPC optimization

The outer-loop MPC adopts the error-augmented LTV model in Eq. (27) as the prediction model. The prediction output corresponds to the reference-aligned tracking error, and the control increment is selected as the optimization variable. This formulation enables explicit incorporation of input-rate constraints and promotes smooth velocity-setpoint generation.

Let the prediction horizon and the control horizon be and , respectively, with . The stacked prediction output and control-increment vectors are defined as

(34)

By recursively propagating the augmented model along the reference trajectory, the prediction relationship is written as

(35)

where and are constructed from , , and along the prediction horizon.

The MPC performance index is selected as

(36)

where and are the weighting matrices for the tracking error and the input increment, respectively. The slack variable is introduced to improve numerical feasibility under constrained operating conditions, and is its penalty coefficient.

Combining Eq. (35) with the input and input-rate constraints gives the following quadratic program:

(37)

The predicted velocity setpoint satisfies

(38)

The constrained receding-horizon formulation in Eq. (37) follows standard stabilizing MPC principles, in which input constraints, input-rate constraints, feasibility, and local closed-loop properties are considered within a unified optimization framework [28,29]. In the present implementation, the same QP structure is used for the periodic MPC benchmark and for the event-triggered outer loop. Their difference lies only in the update schedule of the outer-loop optimization.

After solving the QP, only the first optimal control increment is implemented:

(39)

For compact notation, the MPC solution mapping is denoted by

(40)

where denotes solving the QP in Eq. (37) and extracting the first velocity setpoint.

The MPC prediction model optimizes the nominal setpoint , whereas the physical robot is driven by . Their difference is retained explicitly in the following inner-loop design and practical boundedness analysis.

Inner-loop PID execution and actuator model

The inner loop regulates the actuator response toward the outer-loop MPC setpoint. The velocity tracking error is defined as

(41)

A discrete-time PID-form velocity servo generates the actuator command:

(42)

where , , and are the proportional, integral, and derivative gain matrices, respectively, and is the integral state.

The actuator command is obtained through component-wise physical saturation:

(43)

To mitigate integrator windup, the implementation uses conditional integration rather than a back-calculation gain. For each velocity channel , the integral state is updated as

(44)

The integral state is therefore held whenever the unsaturated command would drive the actuator further into an active saturation limit; otherwise, conventional discrete integration is applied. The reported derivative gains are zero, so the implemented controller is a PI specialization of the PID-form expression in Eq. (42).

The actuator dynamics are modeled as

(45)

where is defined in Eq. (18), and denotes the identity matrix. This model represents the finite response bandwidth of the linear- and angular-velocity channels. Consequently, even when the nominal MPC setpoint satisfies the optimization constraints, the actual actuator velocity can lag behind the setpoint because of actuator dynamics and command saturation.

For the no-PID event-triggered ablation controller, denoted as ET–MPC-same, the command input is directly obtained from the MPC setpoint:

(46)

The same first-order actuator model in Eq. (45) is retained. Therefore, the comparison between ET–MPC-same and ET–MPC–PID isolates the contribution of the fixed-period PID velocity servo under identical nominal MPC settings, triggering parameters, actuator dynamics, and physical constraints.

Event-triggering mechanism

The event-triggering mechanism determines the update instants of the outer-loop MPC. Instead of solving the QP at every sampling instant, the MPC is updated only when the accumulated mismatch relative to the latest optimization instant becomes sufficiently large or when a prescribed maximum holding duration is reached. During each inter-event interval, the outer-loop setpoint is held constant, while the PID–actuator channel continues to operate at the fixed sampling period.

Let denote the most recent outer-loop optimization instant. The triggering vector is defined as

(47)

The first two components describe changes in tracking error relative to :

(48)

where

(49)

The remaining components quantify reference variation:

(50)

and

(51)

Here, is the reference-aligned tracking error defined in Eq. (28). By construction, .

To prevent a single component from dominating the triggering decision solely because of its numerical scale, the normalization vector is defined as

(52)

with normalization matrix

(53)

A diagonal weighting matrix is introduced to adjust the relative importance of the normalized components:

(54)

The scalar normalized triggering metric is

(55)

The basic triggering condition is written as

(56)

where , denoted as in the numerical implementation, is the global normalized trigger threshold. A smaller generally causes more frequent MPC updates and lower holding-induced mismatch, whereas a larger reduces the optimization call rate at the cost of a potentially larger tracking neighborhood.

To avoid both excessively frequent updates and unbounded holding intervals, a minimum inter-event time and maximum holding time are imposed:

(57)

The complete outer-loop update law is

(58)

Whenever an update occurs, . The forced-update condition guarantees that the outer-loop MPC is re-solved within a bounded number of sampling steps even when the normalized triggering metric remains below the threshold.

The selection of and should be coordinated with the response capability of the inner-loop PID–actuator channel. Let denote an equivalent closed-loop response time constant. The velocity tracking error can be approximated by

(59)

where represents the residual contribution of saturation, actuator lag, and unmodeled dynamics. A practical tuning guideline is

(60)

where is the desired decay ratio, is an upper bound on the reference-variation rate, and denotes the allowable holding-induced error.

For reproducibility, is operationally defined as the larger 95% settling time of the linear- and angular-velocity channels in the closed-loop PID–actuator step response. The reference-variation bound is calculated from the sampled reference trajectory as

(61)

The quantity represents a prescribed additional tracking-error budget attributable to retaining an outdated MPC decision. It is used as an engineering tolerance for interpreting the maximum permissible holding interval rather than as an independently optimized controller parameter.

These relations are practical coordination guidelines rather than necessary stability conditions or an unreported analytical parameter-fitting rule. In the reported simulations, they were used as preliminary consistency checks for the inner-loop response and reference variation. The final nominal trigger configuration was selected from the two-parameter sensitivity and Pareto study in Section 4, which explicitly reports the accuracy–computation trade-off.

For the final controller comparison, the same nominal triggering configuration is used for ET–MPC-same and ET–MPC–PID in both trajectories. No auxiliary curvature-trigger term is activated in the sensitivity study or in the final nominal comparison; therefore, the reported update-frequency trade-off is determined directly by and .

External VH–ET–MPC reference implementation

To distinguish the contribution of the proposed fixed-period PID velocity servo from the influence of the event-update policy itself, an external variable-horizon event-triggered MPC reference, denoted as VH–ET–MPC, is implemented following the event-triggered predictive-control principles reported in [24]. This controller is not an ablation of the proposed architecture. Instead, it serves as an external reference method with a distinct event-triggering and stored-sequence execution policy.

At each VH–ET–MPC optimization instant, a constrained MPC problem is solved and the resulting optimal input sequence is stored. Between two optimization events, the stored sequence is advanced and its next element is applied rather than recomputing a new control sequence at every sampling instant. The event condition is based on a normalized state-prediction mismatch between the current closed-loop state and the state predicted from the most recent optimization. The prediction horizon contracts as the stored sequence is executed and is refreshed before the remaining horizon becomes excessively short.

For a same-platform comparison, the VH–ET–MPC reference uses the differential-drive model, sampling period, reference trajectories, QP solver, input bounds, input-rate bounds, and actuator model adopted in this study. It does not include the fixed-period PID velocity servo of the proposed ET–MPC–PID controller. A single normalized mismatch threshold is used for both trajectories. In addition, a minimum dwell time, a bounded open-loop interval, and position and heading error guards are imposed to maintain the comparison within a feasible operating region. The variable-horizon parameters and safeguard values are listed in Table 3.

The VH–ET–MPC reference therefore retains the characteristic prediction-mismatch triggering, stored-input-sequence execution, and contracting-horizon mechanism of the external method, while the plant-side model and physical constraints remain identical to those used for the other MPC-type controllers. This implementation prevents differences in robot model, actuator dynamics, numerical solver, or constraint set from being misinterpreted as differences caused by the event-triggered control policy.

Stability analysis

This subsection provides a local practical boundedness characterization for the proposed ET–MPC–PID controller and its internal no-PID ablation variants. It is not intended as a stability proof for the external VH–ET–MPC reference controller, whose event rule and stored-sequence execution mechanism differ from the proposed structure. Because the proposed event-triggering mechanism holds the MPC setpoint over inter-event intervals and the actual robot input generally differs from the nominal MPC setpoint, strict asymptotic convergence to the origin is not expected in general. The closed-loop tracking error is therefore analyzed in the sense of local ultimate boundedness. The analysis follows local MPC arguments, piecewise-affine representations of constrained MPC, and Lyapunov/BIBO boundedness tools [28–32].

Assumption 1 (Operating region and local prediction validity). The closed-loop trajectory remains in a neighborhood of the reference trajectory in which the error-augmented LTV prediction model in Eq. (27) is valid. Let denote the residual of the local linear approximation. It is assumed that

(62)

for a finite constant throughout this local operating region.

Assumption 2 (Local equivalent feedback form of nominal periodic MPC). Within the operating region, the periodic MPC controller with the QP formulation in Eq. (37) is locally stabilizing. When the active constraint set remains locally constant or switches finitely many times, the constrained MPC law admits a local piecewise-affine state-feedback representation [30,31]. Therefore, the nominal closed-loop error dynamics can be expressed as

(63)

where is Schur stable.

Assumption 3 (Local recursive feasibility). The outer-loop MPC problem is feasible initially and remains feasible within the operating region under standard constrained MPC conditions [28,29]. This is a local assumption consistent with the finite-horizon QP in Eq. (37).

Assumption 4 (Bounded PID–actuator execution mismatch). For the selected PID gains, bounded setpoints, bounded input increments, actuator saturation, and stable first-order actuator dynamics, the local PID–actuator channel has bounded input–bounded output behavior. Consequently, the execution mismatch

(64)

is bounded:

(65)

This condition is consistent with standard BIBO arguments for stable systems with bounded inputs and saturation [32].

Assumption 5 (Bounded holding-induced perturbation). Within the operating region, the reference trajectory and the local MPC solution map are Lipschitz continuous. Under the triggering condition in Eq. (56) and the bounded holding interval in Eq. (57), there exist nonnegative constants and such that the perturbation induced by holding the previous MPC setpoint satisfies

(66)

Under the update law in Eq. (58), any two consecutive optimization instants and satisfy

(67)

The upper bound follows directly from the forced-update condition. Hence, the outer-loop holding duration is bounded, which prevents indefinitely long open-loop execution of an outdated MPC setpoint.

The equivalent perturbation affecting the nominal closed-loop error dynamics is decomposed as

(68)

where is induced by retaining an outdated outer-loop decision, is caused by actuator lag, is caused by command saturation, and is the local linearization residual. Under the preceding assumptions, there exists a finite constant such that

(69)

The perturbed local closed-loop dynamics can therefore be expressed as

(70)

where is the disturbance input matrix. Equation (70) represents the proposed controller as a nominal locally stabilizing MPC loop perturbed by bounded setpoint-holding, actuator-lag, saturation, and local linearization-residual effects.

Since is Schur stable, for any , there exists a unique satisfying [32]

(71)

Choose the Lyapunov candidate

(72)

Combining Eqs. (70) and (71) gives

(73)

Define

(74)

By Young’s inequality, for any ,

(75)

Substituting Eq. (75) into Eq. (73) yields

(76)

Choose and define

(77)

Then

(78)

Using Eq. (69) gives

(79)

Since ,

(80)

Therefore, the Lyapunov function decreases whenever

(81)

The resulting local practical boundedness condition is

(82)

Equation (82) establishes that the closed-loop error converges to a bounded neighborhood whose size depends on the aggregate perturbation . In particular, Eq. (69) clarifies the design trade-off: reducing or decreases the holding-induced component but increases the MPC update frequency; improving the PID–actuator response and avoiding persistent saturation reduce the actuator-related components; and maintaining operation within the local prediction region reduces the contribution of the linearization residual.

For the simulation ablation study, ET–MPC-same uses exactly the same nominal trigger threshold, minimum dwell time, maximum holding time, MPC formulation, constraints, actuator dynamics, and reference trajectory as ET–MPC–PID, while removing only the fixed-period PID velocity servo. ET–MPC-retuned also removes the PID servo but uses more aggressive event-trigger settings to increase the MPC update frequency. These internal comparisons examine whether the benefit of the proposed architecture can be reproduced by increasing MPC update frequency alone, or whether fixed-period velocity compensation remains necessary when the outer-loop optimization is intermittent.

The local practical boundedness result characterizes the deterministic bounded-perturbation model used in the controller analysis. The Monte Carlo study in Section 4 separately evaluates empirical robustness under paired sampled execution uncertainties, including initial-state deviations, actuator-parameter mismatch, measurement noise, and additive equivalent velocity disturbances. This empirical study complements the deterministic local characterization but does not constitute a probabilistic stability guarantee.