Scientific Problem and Ethical Issues

Despite recent advances in passivity-based control, safety standards, and robust actuator tuning, most existing approaches address these aspects in a fragmented manner. In particular, passivity-based control methods, safety constraints, and robust motor control strategies are generally developed independently, thereby limiting their ability to ensure coherent energy safety at the scale of a coupled robotic system. Furthermore, current efforts aimed at raising awareness of the risks associated with increasing robot deployment are not always accompanied by integrated methodological and technical solutions capable of effectively enhancing the safety of automated systems [1, 4]. Consequently, there remains a lack of systemic approaches capable of providing comprehensive energy supervision of a coupled robotic system, encompassing the electric motor, the robot's mechanical structure, and its dynamic interactions with the environment. This scientific gap constitutes the central research problem addressed in the present work.

From both ethical and operational perspectives, the deployment of interactive robots in human-centered environments requires that their physical actions remain continuously supervised and constrained. This necessitates the implementation of explicit energy-supervision mechanisms capable of ensuring that the robot's behavior remains within physically safe limits. In the absence of such mechanisms, uncontrolled energy transfers may lead to hazardous interactions or unexpected behaviors, representing a major concern in current discussions on safety, accountability, and trustworthiness in autonomous and collaborative systems [1, 9, 4, 5, 6].

In this context, two complementary levels of energy safety appear essential:

The integration of these two levels within unified control architecture not only guarantees the dynamic stability and passivity of the overall system but also satisfies the intrinsic safety requirements associated with human-robot interactions.

State of the art

In recent literature, several contributions have partially addressed these issues:

Despite these contributions, there is currently no framework that integrates in a unified manner energy-based modeling, port-Hamiltonian control, supervision through energy tanks, and normative and ethical safety constraints.

Objectives and contributions of the manuscript

This manuscript proposes a unified framework based on the port-Hamiltonian formalism, aimed at ensuring coherent, stable, and safe modeling, control, and energy supervision of coupled motor-robot-environment systems. Unlike existing approaches that address passivity-based control, safety, and energy regulation as separate problems, the proposed framework seeks to bridge several theoretical and methodological gaps related to the global energy supervision of collaborative and intelligent robotic systems.

The main contributions of this work are summarized as follows:

The first layer explicitly limits the power injected by the electric motor in order to guarantee controlled torque transmission. The second layer relies on the analytical sizing of an Energy Tank based on the maximum gravitational torque. This dual-layer structure ensures bounded mechanical energy and safe interactions, even in the presence of uncertainties and external disturbances.

This work highlights the relationship between energy management in robotic systems and data-driven as well as artificial intelligence-based approaches. Although such methods enhance autonomy and decision-making capabilities, they also increase the computational and energetic complexity of robotic platforms. In this context, the proposed port-Hamiltonian framework provides a suitable foundation for ensuring efficiency, robustness, and long-term sustainability in next-generation robotic systems.

The proposed framework guarantees that both injected and stored energy remain physically bounded and continuously supervised. As a result, it contributes to improving the transparency, predictability, and accountability of collaborative robotic systems operating in human-centered environments.

These include experimental validation of the proposed framework, its extension to multi-degree-of-freedom robotic systems, and the integration of machine-learning techniques under passivity constraints to further enhance its applicability, scalability, and practical deployment.

The remainder of this manuscript is organized as follows. Section 2 presents the proposed methodology, detailing the design of the two-level energy-control architecture that integrates motor power regulation and the Energy Tank mechanism within a port-Hamiltonian framework. Section 3 is devoted to the presentation and discussion of the results, including the validation of the Permanent Magnet Synchronous Motor (PMSM) control laws and the evaluation of the energy dynamics of the coupled motor–robot–environment system. Finally, Section 4 concludes the paper and outlines future research perspectives.

Modeling and control of the PMSM for coupling and secure control of robots

The first level of the safety formalism of the industrial robotic system (Fig.1) concerns the control of the Permanent Magnet Synchronous Motor (PMSM). The objective is to impose strictly controlled energy supply, so that the robot evolves according to its intrinsic dynamics, without excess or energy instability. Regardless of the considered robotic architecture-humanoid or industrial-the control of electric actuators plays a central role in two respects. On the one hand, it enables optimization of the overall energy consumption of the system. On the other hand, it ensures precise control of the injected power, whether remotely or in situ, in order to prevent any undesired dynamic response, particularly during interaction with the environment. Thus, the energy regulation of the PMSM constitutes the first link in the overall safety mechanism, ensuring that mechanical-level power exchanges remain compatible with the passivity and stability requirements of the coupled motor-robot-environment system.

Electrical modeling of the PMSM

The considered robotic system is based on the equivalent electrical model of the PMSM expressed in the rotating Park reference frame (d, q). This representation allows a decoupled description of flux and torque dynamics, thereby facilitating the synthesis of control laws adapted to the selected energy-based and passive framework. The electrical model is then written as follows [18]:

where the stator voltage components in the (d,q) reference frame are denoted by U_sd​ and U_sq​, respectively; the permanent-magnet flux linkage per pole is denoted by Ψ_f.

The stator winding resistance is denoted by R_s​; the direct-axis and quadrature-axis inductances are denoted by L_d and L_q​​, respectively; the stator currents in the (d,q) reference frame are denoted by I_sd and I_sq, respectively; the rotational speeds of the rotor and the transmission shaft are denoted by ω_r and ​Ω, respectively; the moment of inertia of the mechanical transmission is denoted by J_m; the viscous friction coefficient of the mechanical transmission is denoted by F_m; and the total resisting torque is denoted by C_r.

The electromagnetic torque of the machine is written as follows [19]:

The expression for instantaneous electrical power is written as:

Synthesis of the PMSM control law

From equation (1), an algebraic rearrangement makes it possible to establish the PMSM current control model, expressed in the form given in equation (4). This formulation highlights the dynamics of the stator currents in the (d, q) reference frame and constitutes the basis for the synthesis of the regulation law.

A further rewriting of equation (4) then leads to the expression of the reference voltages required for converter control, also reported in equation (5). This expression explicitly incorporates the dynamic coupling terms:

where E_d=ω_rL_qI_sq and E_q​=ω_r​(L_d​I_sd​+Ψ_f) represent the cross-coupling compensation components induced by the rotation of the magnetic field. Their consideration ensures effective decoupling of

the dynamics along the direct and quadrature axes.

The introduction of a proportional-integral (PI) controller in equation (5) then makes it possible to establish the open-loop transfer functions (OLTF) and closed-loop transfer functions (CLTF) of the current regulation system. From these expressions, the regulation gains K_p(Isd,Isq) and K_i(Isd,Isq) ​are derived as follows:

Using the parameters in Table A1 in the appendix for the numerical application of the PI gains in equation (6), and for a damping time constant T_r​=1 ms, the following values are obtained: K_i(Isd)​=K_i(Isq)​=958, K_p(Isd​)=5.25, K_p(Isq)=12. These parameters ensure fast and stable current dynamics, compatible with the energy control and safety requirements of the robotic system.

Simulation assumptions for the robust PMSM control law

The assumptions adopted for the implementation of the control law do not in any way alter the physical structure of the motor. They essentially aim to structure the energy exchanges in order to enhance system robustness and preserve its passivity properties.

The objective is to channel the electromagnetic power exclusively along the quadrature axis (q), while ensuring structural passivity along the direct axis (d). This strategy reduces perturbation effects related to cross-couplings between current components and improves the stability of the overall dynamic behavior. The assumptions considered are as follows:

• Vector orientation of the energy flux along the (q) axis:

The vector control principle imposes that the electromagnetic power depends only on the reference current I_sqref. This condition corresponds to opening the energy port on the (q) axis, characterized by U_sqref​=0. Thus, the quadrature component becomes the unique active power transmission channel.

• Cancellation of the reference current on the (d) axis:

Imposing I_sdref​=0 guarantees the absence of active energy contribution on the direct axis. This constraint promotes internal dissipation and stabilizes system dynamics by limiting cross-interactions [18].

• Closure of the energy port on the (d) axis:

Applying the condition U_sd​=0 ensures the passivity of the subsystem associated with the direct axis. This port closure reinforces the decoupling of the (d,q) dynamics, improves robustness with respect to parametric disturbances, and contributes to overall energetic stability.

Under these assumptions, the electromagnetic torque of the PMSM, converted into mechanical power (P_m​), can be expressed as follows [18]:

The validation of the robust modeling associated with the PMSM control law, corresponding to the first level of secure power control, is presented in the Results and Discussion section. This validation is based on three distinct simulation scenarios, designed to evaluate the relevance and robustness of the previously formulated assumptions. These tests make it possible to analyze the dynamic behavior of the system, the effective channeling of power along the (q) axis, as well as the preservation of passivity properties and energetic decoupling.

Port-Hamiltonian method and analysis of the energetic dynamics of a robotic arm

In analytical mechanics, the dynamic modeling of robotic arms mainly relies on energy-based formulations grounded in the notions of kinetic energy and potential energy. Classical approaches thus lead to the Newton-Euler, Euler-Lagrange, and Hamilton equations, each providing a particular level of description of the system's dynamic behavior. Among these formalisms, the Hamiltonian representation constitutes the theoretical foundation of the port-Hamiltonian (PH) paradigm, which extends the classical formulation by explicitly integrating power exchanges through interconnection ports. This framework offers a unified systemic description of energy flows, particularly suited to the analysis and control of coupled electromechanical systems. In the context of a robotic arm, the port-Hamiltonian approach makes it possible to coherently model the energy interactions between the motor, the mechanical structure of the robot, and the environment (human or material). This unification of power interface ports provides a natural framework to [19]:

Thus, energy analysis based on the port-Hamiltonian formalism does not merely constitute an alternative modeling approach, but rather a structuring methodological framework enabling the integration of control, safety, and energy efficiency within a unified approach.

Formalism for energy evaluation and control of a single joint robotic arm

Let us consider a robotic arm as shown in Fig. 2 below, rotating about a jointed axis (0), heavy and rigid. Using the energy provided by the actuator-gear pair (PMSM + gearbox), this arm can be described as follows:

• The distance ℓ is the length of the arm between the rotation axis (0) and the center of gravity of the arm at (G);

• The moment of inertia I represent the resistance to rotation of the arm with respect to axis (0);

• The angle θ or q represents the position or generalized coordinate formed by the arm with the vertical of the inertial frame;

• The gravitational acceleration at G is given by the constant g=9.81m/s²;

• The viscous friction coefficient at the joint is denoted D or B.

Fig.2. Representation of an articulated robotic arm

Determination of the position coordinates and associated energies of the robotic arm

The position coordinates of the arm of length ℓ in the inertial frame are written as:

The positioning velocity components are written as:

The expression of the velocity magnitude is written as:

The expressions of the kinetic energy T(q,q˙​) and the potential energy V(q) of the robotic arm dynamics are written as follows:

Port-Hamiltonian formalism and energetic dynamics of the robotic arm

The port-Hamiltonian formalism applied to the robotic arm is based on a unified energy framework integrating modeling, control, safety, and observation of power exchanges. This approach is structured around the following principles:

• Hamiltonian energy balance:

The dynamics of the robot and its actuator are described from the Hamiltonian H, representing the total energy of the system (kinetic and potential). Its time evolution H’ explicitly reflects the power flows exchanged through the interconnection ports.

• Interconnection and damping assignment-passivity-based control (IDA-PBC):

This strategy makes it possible to act on the interconnection structure and on the dissipation matrices in order to shape a desired Hamiltonian H_d​. The latter is designed as a Lyapunov function, guaranteeing asymptotic stability of the closed-loop system while preserving its passivity properties.

• Energetic safety via the Energy Tank (Etank):

The introduction of an energy tank ensures explicit management of energy exchanges by imposing constraints on the available power. This mechanism prevents any energy overflow that could compromise interaction safety.

• Observation of robot-environment interaction:

The analysis of interconnection ports enables a coherent estimation of the energy flows exchanged with the environment, thus providing a direct indicator of the interaction level and safety.

On this basis, the port-Hamiltonian formalism of the robotic arm is developed in five main steps, allowing the structuring of modeling, control synthesis, and the integration of energetic safety mechanisms.

Step 1: Determination of the actual energy and the energy balance of the robotic arm

Let us first recall the classical expression of the articulated arm dynamics from the Euler-Lagrange formalism as follows [19]:

Where the set of generalized torque couples associated with the joints and the environment is denoted by τ_i; the generalized coordinates of the joints are denoted by q_i​; and the Lagrangian is denoted by L(q,q′) and is given by the following expression:

The Hamiltonian (H), on the other hand, describes the total energy of the robotic arm through the following function:

The energy balance, as a measurable quantity of the Hamiltonian variable, can be written as:

The port-Hamiltonian equations for conceptualizing the robotic arm are defined as follows [11, 19]:

where D represents dissipation or energy losses.

By rearranging Equation (16), the port-Hamiltonian (P-H) state-space representation of the robot dynamics can be expressed as follows:

Furthermore, by identifying the variables and matrices in Equation (17) as: x’=[q_i’; p_i’]^T, [J-R]=[0 I;-I -D]^T, ∇H=[∂H/∂q_i; ∂H/∂p_i]^T, [G]=[0; I]^T and u=[τ_i], the robot dynamics can be rewritten in the following generalized port-Hamiltonian state-space form [19]:

where the matrix (J) represents the interconnection structure matrix, also referred to as the skew-symmetric matrix, which characterizes the internal energy exchanges within the system; the matrix (R) denotes the dissipation matrix, representing the energy losses in the system; the vector (G) corresponds to the port input vector, including the actuators and interactions with the environment; and the variable (u) represents the effort variable applied to the ports, corresponding to the total torques τ_i​.

By rearranging equations (16) and (18) in matrix form and exploiting the properties of symplectic geometry, it is possible to express the variation of the system’s total energy. This variation, identified as follows, allows direct linking of power flows to port actions and dissipation effects [19]:

where the term (∇H^TGu) represents the energy injected into and exchanged through the ports; the term (∇H^TR∇H) represents the dissipated or irreversible internal energy losses; and the variable (τ_i​) denotes the port variable, or total effort applied to the mechanical ports, corresponding to the torque generated by the motor and the interaction with the environment.

From equation (19), the expression for the energy balance of the robotic arm can be written as follows:

where the mechanical torque, or passive port through which the motor injects mechanical power (P_m) into the arm, is denoted by τmreal; the term q′^TDq′ represents the dissipation port (P_d), corresponding to the losses due to viscous friction; and the term q′τenv represents the interaction port power (P_env​), corresponding to the power exchanged between the robotic arm and its environment.

Within the port-Hamiltonian framework, system passivity is guaranteed when the time derivative of the stored energy remains less than or equal to the power exchanged through the system ports, namely.

where y denotes the vector of flow variables associated with (q’). This condition implies that the system cannot generate more energy than it receives from its environment.

The Energy Tank mechanism further strengthens this property by imposing an upper bound on the energy available for control actions. As a result, the energy injected into the system remains limited, preventing excessive accumulation of mechanical energy that could compromise either stability or interaction safety. From a Lyapunov perspective, the augmented total energy (H + Etank) can be regarded as a positive Lyapunov candidate function whose time evolution remains bounded and dissipative. Consequently, the Energy Tank serves not only as an energy-protection mechanism but also as a means of enhancing system stability by preserving passivity and preventing any uncontrolled amplification of energy exchanges.

Step 2: IDA-PBC control law of the Port-Hamiltonian system

Let us rewrite the Hamiltonian H(p,q) of the studied robotic arm as follows:

Let us recall the real port-Hamiltonian formulation in terms of p and q as follows:

The objective of the IDA-PBC (Interconnection and Damping Assignment Passivity-Based Control) method is to shape the energy dynamics of the system by imposing a desired port-Hamiltonian structure together with a desired Hamiltonian function (Hd), which are respectively defined as follows [14, 20, 21]:

where K_d is the derivative damping gain.

where K_p is the virtual stiffness constant or virtual damping proportional gain, and q_d is the desired generalized coordinate.

It should be emphasized that Equations (24) and (25) are not obtained as a direct mathematical consequence of the preceding equations. Rather, they arise from a design choice inherent to the IDA-PBC methodology, which is based on the principle of energy shaping. More specifically, their formulation is inspired by the structure of the actual port-Hamiltonian system given by Equation (16) and by the expression of the actual Hamiltonian defined in Equation (14). The objective is to define a desired Hamiltonian (H_d), interpreted as a candidate Lyapunov function, in order to shape the dynamics of the robotic manipulator around the desired equilibrium configuration (q_d), corresponding to the strict minimum of the desired energy.

By subsequently substituting the expression of the desired Hamiltonian given in Equation (25) into the desired port-Hamiltonian formulation of Equation (24), the following expression of the desired port-Hamiltonian system is obtained:

The search for the IDA-PBC control law on the port torques τ_i consists of ensuring H′_d ≤ 0 as a Lyapunov function. To do this, we identify p′ in equations (23 and 26) and deduce the expression of the ports associated with the desired energy as follows [14, 20]:

According to the fundamental theorem of rotational dynamics, we can write:

Equation (28) of the closed-loop dynamics corresponds to the canonical form of the following second-order linear dynamics:

where the natural undamped frequency is denoted by ω_n; the damping coefficient is denoted by ξ; and the stiffness constant of the energy well is denoted by K_p. By identifying equations (28) and (29), we can write:

The derivations based on Equations (28), (29), and (30) show that the gain K_d is determined from the desired closed-loop dynamics, ensuring a trade-off between response speed and oscillation damping. Consequently, the value of K_d is directly related to the damping ratio ξ, thereby guaranteeing controlled energy dissipation within the system.

In contrast, the Energy Tank parameters are defined from the maximum gravitational torque τ_g^max, together with an additional safety margin to ensure passivity and to bound the energy exchanged during operation. These design choices directly affect system stability, torque-limitation capabilities, and robustness against external disturbances.

Compared with conventional control approaches such as PI and LQR controllers, which are primarily focused on reducing tracking errors, the proposed method introduces explicit supervision of energy flows through the combination of the port-Hamiltonian framework and an Energy Tank mechanism. This architecture enables energy constraints to be directly embedded into the control law, thereby simultaneously guaranteeing system stability, passivity, and physical safety.

Recent studies reported in the literature further confirm that energy-based control approaches provide stronger guarantees for robot-environment interactions, particularly in collaborative robotics applications where safety, robustness, and predictable behavior are critical requirements.

The choices of ω_n ϵ[10 to 30] rad/s and ξ = 0.023, corresponding to the desired industrial arm, and using the parameter values from Table A1 in the appendix, allow determining the following K_d and K_p:

The simulation model of the actually desired mechanical torque, associated with the energy tank for safety, obeys the formalism of the IDA-PBC/Energy Tank law as follows [8, 11, 13, 20]:

To guarantee passivity and energy safety of the ports, the energy tank (E_tank / τ_tank^max) serves to limit the injection of motor energy, by considering either the calculated or desired real torque (τ_i^cal) if E_tank > E_min; otherwise, by setting the damping torque (τ_tank^max / α τ_i^cal).

The choice of the tank energy (E_tank), in accordance with ISO-type standards in robotics [8, 21, 22], allows writing:

where τ_g^max represents the maximum gravitational torque.

In practice, the gravitational torque (τ_g) corresponds to the gradient of potential energy and takes into account both:

• the local peak or maximum static load at the joint;

• and the minimum energy required to lift the robot’s mechanical link.

Meanwhile, the Energy Tank (Etank) acts as an energy-supervision mechanism designed to limit the amount of energy that can be injected into the robot-environment interaction. Its sizing is based on the maximum gravitational torque that may be exerted at the robot joint. This torque is defined as τ_g^max​=mgℓ and corresponds to the worst-case static load that the system must be able to compensate.

To introduce a safety margin ensuring both system passivity and the robustness of energy exchanges, the tank-sizing coefficient is selected within the safety interval (1.5, 2), as specified in Equation (33). In the present study, the most conservative choice is adopted by setting this coefficient equal to 2. The maximum admissible torque associated with the Energy Tank is therefore defined as:

The factor 2 is a dimensionless safety coefficient and should not be interpreted as a torque value under any circumstances. Consequently, the quantity τ_tank^max​(N.m) represents the only torque bound employed in the proposed energy-supervision mechanism. It defines the upper limit of the energy exchanges authorized by the control law and ensures that the available mechanical energy remains compatible with the passivity, stability, and safety requirements of the robot-environment system.

For the studied robotic arm, whose joints are rigid and subject to gravity, these considerations allow establishing the following relations [8, 21, 22]:

Using equation (35), it is possible to determine the maximum tank torque (τ_tank) not to be exceeded in the studied model. Hence, the maximum gravitational torque τ_g^max is reached for |sinq| = 1, which leads to: τ_g^max = m g ℓ = 12 * 9.81 * 0.6 = 70.632 N.m.

From equation (34), the maximum energy tank torque is then calculated as follows: τ_tank = 2 * τ_g^max = 141.264 N.m.

This value (τ_tank) defines the available power limit in the tank, ensuring that the port-Hamiltonian control remains passive and safe, while covering the maximum energy requirements of the robotic arm during operations under gravitational load.

Third step: Validation of safety bounds and local performance of the torque model

The objective of this step is twofold:

• verify energy safety: demonstrating that, regardless of variations in energy exchanges, the torque or effective energy remains bounded by the global safety limit defined by the tank (τ_tank^max);

• evaluate local performance: analysis of the accuracy of the exact torque model around the desired set point, in order to guarantee dynamic behavior consistent with control objectives.

For this, the following parameters are used in equation (27) of the torque:

D - K_d = -3.07 N.m, mgℓ = 70.632 N·m²/s², K_p = 672

This leads to the calculated torque expression: τ_cal = 70.632 sin(q) - 672 (q - q_d) - 94.07 q′

The simulation assumptions adopted are as follows:

• Four desired positions q_d, representative of physically significant configurations: q_d = (0°, 0 rad); (30°, 0.524 rad); (45°, 0.7854 rad); (90°, 1.571 rad);

• The arm is initially at rest, q′ = 0°, at critical static instants;

• The maximum tank limit is τ_tank^max = 2 * |τ_g| = 141.264 N.m.

This configuration allows verification that the calculated torque τ_cal remains strictly below the safety limit τ_tank^max while evaluating the local fidelity of the model with respect to the setpoint q_d.

a) Validation of Etank safety bound

The static equilibrium condition corresponds to:

∆q = q - q_d = 0 and q = q_d

The calculated torque values τcal for the chosen setpoint angles are as follows:

τ_cal (0) = 0 N.m; τ_cal (30°) = 35.316 N.m; τ_cal (45°) = 49.944 N.m; τ_cal (90°) = 70.632 N.m

It is observed that for any desired angle q_d ϵ [-180°, 180°], the calculated torque systematically respects the conditions: τ_cal < τ_tank and τ_cal = τ_real

Therefore, the energy tank E_tank is always respected, and no exceedance of the maximum torque is observed. This constitutes proof of global safety, independent of the exact configuration of the robotic arm, and guarantees local passivity, even for high K_p gains.

b) Validation of local performance of the exact model

The realism allows considering error variation choices such that ∆q is: (±3°, ±0.05 rad); (±6°, ±0.10 rad); (±7°, ±0.122 rad). Additionally, q ≠ q_d and q′ = 0.

In this validation case, we consider the locally critical desired position q_d = 90°. Therefore, q = ∆q + 90°, and we can write: τ_cal = 70.632 cos (∆q) - 672 * ∆q

The values of τ_cal (90°) corresponding to the following ∆q errors are:

• For ∆q (+0.05, -0.05) rad: τ_cal = 36.935 N.m, 104.135 N.m;

• For ∆q (+0.10, -0.10) rad: τ_cal = 3.045 N.m, 137.45 N.m;

• For ∆q (+0.122, -0.122) rad: τ_cal = -11.878 N.m, 152.1 N.m.

Unlike the static equilibrium study, the exact model reaches and exceeds the saturation performance τ_tank = 141.264 N.m, when at the critical desired state q_d = 90° and for ∆q = -0.122 rad. Therefore, we see that τ_cal = 152.1 N.m is greater than τ_tank = 141.264 N·m.

Consequently, we can write: τ_real = τ_tank or τ_real = α τ_cal; to signify that the sizing is consistent with the energy tank conditions of equation (32).

Modeling of the complete motor-robot-environment dynamics

The modeling of the complete energy dynamics allows coupling and evaluating the motor-robot-environment interaction, such that:

This novel complete coupling model at the robotic arm joint unifies the ports of energy torques that interact with the joint efficiency and the motor’s controlled mechanical speed.

Consider the Hamiltonian passivity equation at the joint as follows:

The classical reduction ratio at the joint is N.

The motor-side energy projection and conservation of power at the joint allow determining the efficiency without dissipation losses as follows:

where the mechanical torque injected and current-controlled is denoted by τm; the term (τ_real/Nη) represents the torque used to move the robot; and the term (τ_env/Nη) represents the torque exchanged with the environment.

The coupled motor-robot-environment dynamics can be compactly expressed as a residual-torque-driven motor equation, ensuring physical causality and energy consistency. The torque dynamics equation can be written as:

where, small(s) is the Laplace operator (s=jw).

To determine the complete coupling, equation (37) is introduced into equation (38) of the energy transmission dynamics, yielding the following complete coupling model:

The coupled motor-robot-environment dynamics can be compactly expressed as a residual-torque-driven motor equation, ensuring physical causality and energy consistency.

Validation test of the assumptions of the robust modeling of the PMSM control law

The validation of the motor control law assumptions consists of verifying the following tests by simulation:

The quadrature reference current I_sq-ref is a ramp varying from 0 A to 5 A, respectively from 0 s to 30 s and from 30 s to 60 s. In addition, we take a current I_sd-ref = 5 A (I_sd-ref ≠ 0) and U_sd ≠ 0. The simulation results of Fig. 3 are obtained. Under these conditions, it is observed that the current I_sd in Fig. 3a generally follows its reference but undergoes disturbances after each transient regime at times 0 s and 30 s. Moreover, in Fig. 3b, the current I_sq also follows its reference but exhibits a deviation or tracking error with respect to I_sq-ref. These observations indicate that opening the ports (I_sd-ref ≠ 0 and U_sd ≠ 0) increases the sensitivity of the control model to disturbances and reference deviations, highlighting the importance of closing the direct port to ensure the robustness and passivity of the control.

The quadrature reference current I_sq-ref is defined as a ramp, identical to that of the first test. The direct-axis reference current is set to I_sd-ref = 0 A while the voltage U_sd ≠ 0 V. The simulation results are presented in Fig. 4.

In this second test, it is observed that the current I_sd in Fig. 4a generally follows its reference but undergoes disturbances after each transient regime, at times 0 s and 30 s. Meanwhile, in Fig. 4b, the current I_sq follows its reference with a very small deviation, indicating good tracking despite the presence of disturbances.

These results show that opening the direct port (U_sd ≠ 0 V) leads to increased sensitivity of the control model to disturbances and reference deviations, although the errors remain small on the quadrature axis. This analysis confirms that closing the direct port is essential to maximize robustness and passivity of the control, by minimizing the influence of disturbances on the current dynamics.

The quadrature reference current I_sq-ref is this time defined as a decreasing ramp, from 5 A to 1 A, respectively over the intervals from 0 s to 30 s and from 30 s to 60 s. The direct-axis reference current is set to I_sd-ref= 0 A and the direct voltage U_sd= 0 V. The simulation results are presented in Fig. 5.

The current I_sd in Fig. 5a perfectly follows its reference, without disturbance or accuracy error. In addition, in Fig. 5a, the current I_sq also follows its reference, without significant deviation and with almost zero fluctuation.

These results demonstrate that complete port closure (I_sd-ref = 0 A and U_sd = 0 V) eliminates the sensitivity of the control model to disturbances and reference deviations. Thus, this test fully validates the port closure assumption, confirming that the PMSM control law is robust to external disturbances and model variations, and guarantees accurate and stable current tracking.

As a preliminary step to the formulation of the hypotheses underlying the robust control law of the Permanent Magnet Synchronous Motor (PMSM) presented in Section 2.1.3, three test configurations were carried out to assess the robustness of different port opening and closing conditions, with the aim of selecting the most appropriate control model for the system.

In summary, the obtained results lead to the following observations:

This case shows that, under the influence of coupling effects represented by the terms (E_d) and (E_q) in Equation (5), the control model exhibits sensitivity to disturbances induced by abrupt changes in reference commands (I_sdref = 5 A at t = 0 s, Fig. 3a, and I_sqref = 5 A at t = 30 s, Fig. 3b). This configuration also leads to a steady-state tracking error between the reference I_sqref and the actual current (I_sq), characterized by a relative error of (E_r = 2.4%) as shown in Fig. 3b.

In this configuration, the results show that a sudden variation in I_sq (Fig. 4b at t = 30s) still induces a disturbance on the I_sd component (Fig. 4a at t = 30s), indicating that coupling effects (E_d) and (E_q) remain present. Moreover, the non-zero voltage port U_sd ≠ 0 introduces additional fluctuations in I_sd (Fig. 4a) and contributes to a tracking error between I_sqref and I_sq, with a relative error of (E_r = 0.1%) observed in Fig. 4b.

In contrast to the previous configurations, this case shows that abrupt variations in I_sq induce almost no disturbance on I_sd (Fig. 4a, t = 0s and t = 30s), indicating a significant attenuation of coupling effects associated with (E_d) and (E_q). Furthermore, simultaneous port closure leads to a reduction in tracking error, with a relative error of approximately (E_r ≈ 0%) observed in Figs. 5a and 5b.

Based on this comparative analysis, the configuration (I_sd = 0 and U_sd = 0) is retained for the subsequent development of the robust PMSM control law. This choice makes it possible to reduce cross-coupling between the d- and q-axes, improve the robustness of the motor’s dynamic behavior, and ensure better consistency with the passivity and energy-control requirements of the overall motor-robot-environment system.

Simulated evaluation of the energy dynamics of the second safety level of a complete coupling model (motor, robot and environment)

The simulated evaluation and validation of the secure energy dynamics of the complete coupling system (motor, robot and environment) are carried out through three distinct scenarios:

In all simulation cases, the desired robot position is initially fixed at maximum effort, q_d = pi/2. The control of the power supplied by the motor is ensured by:

Figs.5a and 5b illustrate that the power injected by the motor can be controlled precisely and optimally for the robot and, possibly, for its environment. This approach constitutes the first level of energy safety, allowing the human operator to maintain direct control over the energy transmitted to an industrial or humanoid robot.

The second level of energy safety is then evaluated through the three mentioned cases, based on the sizing and simulation of the energy tank Etank, in order to analyze the global energy dynamics of the robot under different load and gravitational compensation conditions.