Consider a class of MRMs consisting of $$ n $$ modules. Each self-contained rotary joint module integrates the following components: a direct current (DC) motor serving as the actuator, and two encoders located on the motor side and the link side, respectively, for acquiring the position and velocity information of both the motor and the link. A harmonic drive is employed as a speed reducer. The angular positions at the harmonic drive components, as detailed in ^[15], can be determined using the ideal input–output kinematic relationship.

(1) $$ \begin{equation} {{\theta }_{wi}}={{\gamma }_{i}}{{\theta }_{fi}} \end{equation} $$

where subscript “$$ i $$” is $$ i $$th module. Denote wave-generator angular position as $$ {{\theta }_{wi}} $$, gear ratio as $$ {{\gamma }_{i}} $$, flexspline output position as $$ {{\theta }_{fi}} $$. Ideal static force equilibrium among components can then express as

(2) $$ \begin{equation} {{\tau }_{wi}}=\frac{1}{{{\gamma }_{i}}}{{\tau }_{fi}} \end{equation} $$

where the variables $$ {{\tau }_{wi}} $$ and $$ {{\tau }_{fi}} $$ signify torque acting on the wave generator as well as the flexspline output torque, respectively. As is customary in the existing literature and standard practice, the circular spline is held stationary, thereby designating the wave generator as the input member and the flexspline as the output. Consequently, the relationship $$ {{\theta }_{ci}}=0 $$ holds, and the torque associated with the fixed circular spline requires no further consideration.

Nevertheless, empirical data characterizing the input–output relationship exhibit a pronounced departure from linearity; specifically, the output does not scale proportionally with the input. Potential sources of this nonlinear behavior include frictional torques, torsional compliance within the harmonic drive components, and kinematic inaccuracies. By first establishing the ideal kinematic constraints that govern motion and force transmission in a harmonic drive, these additional dynamic effects can be systematically incorporated through the introduction of friction, compliance, and kinematic error terms. The resulting compliant response of an MRM joint actuated by a harmonic drive is considered, which references the established compliance model of HD. $$ {{\theta }_{wOi}} $$ and $$ {{\theta }_{wIi}} $$ represent angular positions of wave generator's outer as well as central parts within the joint, while $$ {{\theta }_{fOi}} $$ as well as $$ {{\theta }_{fIi}} $$ mean angular position of flexspline at the load and gear side. The quantity $$ {{\tau }_{mri}} $$ means motor friction, and $$ {{\tau }_{fri}} $$ denotes lumped friction torque of harmonic drive, encompassing bearing friction in the wave generator, friction arising from gear-tooth meshing between the flexspline and circular spline, in addition to output bearing friction, as perceived from the output side of the transmission. Furthermore, $$ {{I}_{mi}} $$ denotes combined inertia of DC motor, while $$ {{k}_{wi}} $$ as well as $$ {{k}_{fi}} $$ are local elastic coefficients for wave generator as well as flexspline.

Upon incorporating frictional losses within harmonic drive transmission, Equation (2) transforms into the following form

(3) $$ \begin{equation} {{\tau }_{wi}}={{\tau }_{i}}-{{\tau }_{mri}}=\frac{1}{{{\gamma }_{i}}}({{\tau }_{fi}}-{{\tau }_{fri}}) \end{equation} $$

where $$ {{\tau }_{i}} $$ is motor torque. Furthermore, the torsional deformations of the flexspline as well as the wave generator can be defined

(4) $$ \begin{equation} \Delta {{\theta }_{fi}}={{\theta }_{fOi}}-{{\theta }_{fIi}} \end{equation} $$

(5) $$ \begin{equation} \Delta {{\theta }_{wi}}={{\theta }_{wOi}}-{{\theta }_{wIi}} \end{equation} $$

It is worth noting from Equations (4) and (5) that only the wave generator input position $$ {{\theta }_{wIi}} $$ (i.e., the motor angle) and the flexible output position $$ {{\theta }_{fOi}} $$ (i.e., the joint angle) can be measured, position measurements are acquired via the motor-side encoder as well as the link-side encoder^[21,22]. The harmonic drive’s total torsional angle within the $$ i $$th joint may subsequently be determined from these available positions using the relationship presented below

(6) $$ \begin{equation} \Delta {{\theta }_{i}}={{\theta }_{fOi}}-\frac{{{\theta }_{wIi}}}{{{\gamma }_{i}}} \end{equation} $$

By adding and subtraction $$ {{\theta }_{fIi}} $$ as well as $$ \frac{{{\theta }_{wOi}}}{{{\gamma }_{i}}} $$ in Equation (6), it has

(7) $$ \begin{equation} \begin{aligned} & \Delta {{\theta }_{i}}={{\theta }_{fOi}}-\frac{{{\theta }_{wIi}}}{{{\gamma }_{i}}}\Delta {{\theta }_{i}}={{\theta }_{fOi}}-{{\theta }_{fIi}}+\left( \frac{{{\theta }_{wOi}}}{{{\gamma }_{i}}}-\frac{{{\theta }_{wIi}}}{{{\gamma }_{i}}} \right)+\left( {{\theta }_{fIi}}-\frac{{{\theta }_{wOi}}}{{{\gamma }_{i}}} \right) \\ & =\Delta {{\theta }_{fi}}+\frac{\Delta {{\theta }_{wi}}}{{{\gamma }_{i}}}+\Delta {{\theta }_{upi}} \end{aligned} \end{equation} $$

where $$ \Delta {{\theta }_{upi}} $$ represents kinematic error, defined as difference between measured flexspline output as well as its expected value, and is expressed as

(8) $$ \begin{equation} \Delta {{\theta }_{upi}}={{\theta }_{fIi}}-\frac{{{\theta }_{wOi}}}{{{\gamma }_{i}}} \end{equation} $$

As reported in the literature, achieving higher accuracy in joint torque estimation necessitates accounting for the torque transmission deviation and kinematic error of the harmonic drive, and establishing an error model to compensate for these inaccuracies^[23]. The error model is defined as follows:

(9) $$ \begin{equation} \left\{ \begin{aligned} & \Delta {{\theta }_{upi}}={{{\tilde{\theta }}}_{wi}}+{{{\tilde{\theta }}}_{fi}} \\ & {{{\tilde{\theta }}}_{wi}}={{a}_{w0}}+{{a}_{w1}}\cos ({{\omega }_{w}}{{\theta }_{wIi}})+{{b}_{w1}}\sin ({{\omega }_{w}}{{\theta }_{wIi}}) \\ & +{{a}_{w2}}\cos (2{{\omega }_{w}}{{\theta }_{wIi}})+{{b}_{w2}}\sin (2{{\omega }_{w}}{{\theta }_{wIi}}) \\ & {{{\tilde{\theta }}}_{fi}}={{a}_{f0}}+{{a}_{f1}}\cos ({{\omega }_{f}}{{\theta }_{fOi}})+{{b}_{f1}}\sin ({{\omega }_{f}}{{\theta }_{fOi}}) \\ \end{aligned} \right. \end{equation} $$

where $$ {{\tilde{\theta }}_{wi}} $$ as well as $$ {{\tilde{\theta }}_{fi}} $$ denote kinematic errors corresponding to wave generator and flexspline. The determined parameters are summarized in Table 1.

Given that the typical stiffness and hysteretic characteristics of an HD, as reported in the literatur^[24], indicate that the local elastic coefficient increases with the flexspline torque, this relationship governs the following definition of the coefficient:

(10) $$ \begin{equation} {{k}_{fi}}=\frac{d{{\tau }_{fi}}}{d\Delta {{\theta }_{fi}}} \end{equation} $$

Given the symmetry inherent in the stiffness characteristics of HD, local elastic coefficient can be approximated as

(11) $$ \begin{equation} {{k}_{fi}}\text{=}{{k}_{fi\text{0}}}\left( 1+{{({{c}_{fi}}{{\tau }_{fi}})}^{2}} \right) \end{equation} $$

where $$ {{k}_{fi\text{0}}} $$ as well as $$ {{c}_{fi}} $$ mean certain constants. If $$ {{k}_{fi\text{0}}}\ne \text{0} $$, then flexspline torsion is

(12) $$ \begin{equation} \Delta {{\theta }_{fi}}\text{=}\int_{\text{0}}^{{{\tau }_{fi}}}{\frac{d{{\tau }_{fi}}}{{{k}_{fi}}}}\text{=}\frac{\arctan ({{c}_{fi}}{{\tau }_{fi}})}{{{c}_{fi}}{{k}_{fi0}}} \end{equation} $$

Moreover, the deformation range of the harmonic drive contracts sharply, approaching zero at the rated torque, which indicates a pronounced increase in the stiffness of the wave generator. To characterize the hysteretic behavior arising from this stiffness profile, the local elastic coefficient of the wave generator is expressed as follows:

(13) $$ \begin{equation} {{k}_{wi}}={{k}_{wi0}}{{e}^{{{c}_{wi}}\left| {{\tau }_{wi}} \right|}} \end{equation} $$

where $$ {{k}_{wi0}} $$ as well as $$ {{c}_{wi}} $$ mean certain constants. If $$ {{k}_{wi\text{0}}}\ne \text{0} $$, $$ \Delta {{\theta }_{wi}} $$ is determined via relationship

(14) $$ \begin{equation} \Delta {{\theta }_{wi}}\text{=}\int_{\text{0}}^{{{\tau }_{wi}}}{\frac{d{{\tau }_{wi}}}{{{k}_{wi}}}}\text{=}\frac{sgn ({{\tau }_{wi}})}{{{c}_{wi}}{{k}_{wi0}}}(1-{{e}^{-{{c}_{wi}}\left| {{\tau }_{wi}} \right|}}) \end{equation} $$

where $$ sgn (\cdot ) $$ is

(15) $$ \begin{equation} sgn ({{\tau }_{wi}})=\left\{ \begin{matrix} 1 & {} & {{\tau }_{wi}}>0 \\ 0 & {} & {{\tau }_{wi}}=0 \\ -1 & {} & {{\tau }_{wi}}<0 \\ \end{matrix} \right. \end{equation} $$

Thus, the overall torsional angle of the harmonic drive is derived by substituting Equations (12) and (14) into Equation (7).

(16) $$ \begin{equation} \begin{aligned} & \Delta {{\theta }_{i}}=\Delta {{\theta }_{fi}}+\frac{\Delta {{\theta }_{wi}}}{{{\gamma }_{i}}}+\Delta {{\theta }_{upi}} \\ & \text{=}\frac{\arctan ({{c}_{fi}}{{\tau }_{fi}})}{{{c}_{fi}}{{k}_{fi0}}}\text{+}\frac{sgn ({{\tau }_{wi}})}{{{\gamma }_{i}}{{c}_{wi}}{{k}_{wi0}}}(1-{{e}^{-{{c}_{wi}}\left| {{\tau }_{wi}} \right|}})+\Delta {{\theta }_{upi}} \end{aligned} \end{equation} $$

Substitution of measured link-side and motor-side position values into Equation (6) provides HD torsional angle $$ \Delta {{\theta }_{i}} $$. Upon rearrangement of the foregoing equation, the corresponding joint torque, which constitutes the flexspline output of the harmonic drive, is estimated according to the following relation

(17) $$ \begin{equation} \begin{aligned} & {{\tau }_{fi}}\text{=}{{\tau }_{fci}}-{{\tau }_{upi}} \\ & {{\tau }_{fci}}=\frac{1}{{{c}_{fi}}}\tan ({{c}_{fi}}{{k}_{fi0}}(\Delta {{\theta }_{i}}-\frac{sgn ({{\tau }_{wi}})}{{{\gamma }_{i}}{{c}_{wi}}{{k}_{wi0}}}(1-{{e}^{-{{c}_{wi}}\left| {{\tau }_{wi}} \right|}}))) \\ \end{aligned} \end{equation} $$

where $$ {{\tau }_{fci}} $$ represents the deterministic component of estimated joint torque. The output torque ripple $$ {{\tau }_{upi}} $$, regarded as a model uncertainty term, can be addressed in the compensation scheme described in the following section. Meanwhile, $$ {{\tau }_{wi}} $$ is approximated using the motor torque.

Based on the robot modeling method with torque feedback technique reported in the literature, the dynamics of the MRM system are formulated as a synthesis of individual joint subsystems. Among them, the dynamic model of the $$ i $$th subsystem is expressed as

(18) $$ \begin{equation} {{I}_{mi}}{{\gamma }_{i}}{{\ddot{q}}_{i}}+{{f}_{i}}({{q}_{i}},{{\dot{q}}_{i}})+{{Z}_{i}}(q,\dot{q},\ddot{q})+\frac{{{\tau }_{fci}}}{{{\gamma }_{i}}}+{{d}_{i}}({{q}_{i}})={{\tau }_{i}} \end{equation} $$

$$ {{I}_{mi}} $$ represents moment of inertia of the motor; $$ {{\gamma }_{i}} $$ denotes gear ratio; $$ q $$ represents position of joints; $$ {{f}_{i}}({{q}_{i}},{{\dot{q}}_{i}}) $$ denotes friction torque; $$ {{\tau }_{i}} $$ indicates motor output torque; $$ {{Z}_{i}}(q,\dot{q},\ddot{q}) $$ represents the IDC between modules and $$ {{d}_{i}}({{q}_{i}}) $$ represents the torque transmission error of HD and measuring error.

We consider that the friction term $$ {{f}_{i}}({{q}_{i}},{{\dot{q}}_{i}}) $$ encompasses the concentrated friction of the harmonic drive as well as motor friction. Based on references, the frictional term is described

(19) $$ \begin{equation} {{f}_{i}}({{q}_{i}},{{\dot{q}}_{i}})={{b}_{fi}}{{\dot{q}}_{i}}+\left( {{f}_{ci}}+{{f}_{si}}{{e}^{(-{{f}_{\tau i}}\dot{q}_{i}^{2})}} \right)sgn ({{\dot{q}}_{i}})+{{f}_{qi}}({{q}_{i}},{{\dot{q}}_{i}}) \end{equation} $$

where $$ {{b}_{fi}} $$$$ {{f}_{ci}} $$$$ {{f}_{si}} $$$$ {{f}_{\tau i}} $$ denote friction model parameter, $$ {{f}_{qi}}({{q}_{i}},{{\dot{q}}_{i}}) $$ is position dependent friction as well as other friction of modeling errors. Based on the linearization strategy presented in the reference^[25], suppose that $$ {{f}_{si}} $$ and $$ {{f}_{\tau i}} $$ are close to their actual values. Therefore, $$ {{f}_{si}}{{e}^{(-{{f}_{\tau i}}\dot{q}_{i}^{2})}} $$ can be linearized. So that the higher-order terms of $$ {{f}_{si}}{{e}^{(-{{f}_{\tau i}}\dot{q}_{i}^{2})}} $$ can be ignored as follows

(20) $$ \begin{equation} \begin{aligned} & {{f}_{si}}{{e}^{({{f}_{\tau i}}\dot{q}_{i}^{2})}}\text{=(}{{{\hat{f}}}_{si}}+{{{\tilde{f}}}_{si}}\text{)(}{{e}^{(-{{{\hat{f}}}_{\tau i}}\dot{q}_{i}^{2})}}-{{{\dot{q}}}^{2}}_{i}{{{\tilde{f}}}_{\tau i}}{{e}^{(-{{{\hat{f}}}_{\tau i}}\dot{q}_{i}^{2})}}\text{)} \\ & \approx {{{\hat{f}}}_{si}}{{e}^{(-{{{\hat{f}}}_{\tau i}}\dot{q}_{i}^{2})}}+{{{\tilde{f}}}_{si}}{{e}^{(-{{{\hat{f}}}_{\tau i}}\dot{q}_{i}^{2})}}-{{{\dot{q}}}^{2}}_{i}{{{\tilde{f}}}_{\tau i}}{{{\hat{f}}}_{si}}{{e}^{(-{{{\hat{f}}}_{\tau i}}\dot{q}_{i}^{2})}} \end{aligned} \end{equation} $$

Substituting Equation (20) into Equation (19), $$ {{f}_{i}}({{q}_{i}},{{\dot{q}}_{i}}) $$ is approximated as:

(21) $$ \begin{equation} {{f}_{i}}({{q}_{i}},{{\dot{q}}_{i}})\approx {{\hat{b}}_{fi}}{{\dot{q}}_{i}}+\left( {{{\hat{f}}}_{ci}}+{{{\hat{f}}}_{si}}{{e}^{({{{\hat{f}}}_{\tau i}}\dot{q}_{i}^{2})}} \right)sgn ({{\dot{q}}_{i}})+{{f}_{qi}}({{q}_{i}},{{\dot{q}}_{i}})+Y({{\dot{q}}_{i}}){{\tilde{F}}_{i}} \end{equation} $$

where $$ {{\tilde{F}}_{i}}={{\left[ \begin{matrix} {{b}_{fi}}-{{{\hat{b}}}_{fi}} & {{f}_{ci}}-{{{\hat{f}}}_{ci}} & {{f}_{si}}-{{{\hat{f}}}_{si}} & {{f}_{\tau i}}-{{{\hat{f}}}_{\tau i}} \\ \end{matrix} \right]}^{T}} $$ represents parameter uncertainty of frictional. $$ {{\hat{b}}_{fi}},{{\hat{f}}_{ci}},{{\hat{f}}_{si}} $$ and $$ {{\hat{f}}_{\tau i}} $$ denote the estimated value of friction parameters.

$$ Y({{\dot{q}}_{i}}) $$ is

(22) $$ \begin{equation} Y({{\dot{q}}_{i}})=\left[ \begin{matrix} {{{\dot{q}}}_{i}} & sgn ({{{\dot{q}}}_{i}}) & {{e}^{(-{{{\hat{f}}}_{\tau i}}{{{\dot{q}}}_{i}})}}sgn ({{{\dot{q}}}_{i}}) & -{{{\hat{f}}}_{si}}\dot{q}_{i}^{2}{{e}^{(-{{{\hat{f}}}_{\tau i}}{{{\dot{q}}}_{i}})}}sgn ({{{\dot{q}}}_{i}}) \\ \end{matrix} \right] \end{equation} $$

Term $$ {{Z}_{i}}(q,\dot{q},\ddot{q}) $$ is produced by the coupling effect of the bottom joint module, which is defined

(23) $$ \begin{equation} {{Z}_{i}}(q,\dot{q},\ddot{q})={{I}_{mi}}\sum\nolimits_{j=1}^{i-1}{z_{mi}^{T}{{z}_{qj}}{{{\ddot{q}}}_{j}}}+{{I}_{mi}}\sum\nolimits_{j=2}^{i-1}{\sum\nolimits_{k=1}^{j-1}{z_{mi}^{T}\left( {{z}_{qk}}\times {{z}_{qj}} \right){{{\dot{q}}}_{k}}{{{\dot{q}}}_{j}}}} \end{equation} $$

where $$ {{z}_{mi}} $$$$ {{z}_{qj}} $$$$ {{z}_{qk}} $$ are unity vectors along axis of rotation of $$ i $$th rotor, $$ j $$th joint as well as $$ k $$th joint. For facilitating the analysis of IDC between modules, according to the theory in the literature^[13], $$ {{I}_{mi}}\sum\limits_{j=1}^{i-1}{z_{mi}^{T}{{z}_{qj}}{{{\ddot{q}}}_{j}}} $$ and $$ {{I}_{mi}}\sum\limits_{j=2}^{i-1}{\sum\limits_{k=1}^{j-1}{z_{mi}^{T}\left( {{z}_{qk}}\times {{z}_{qj}} \right)}{{{\dot{q}}}_{k}}{{{\dot{q}}}_{j}}} $$ can be rewritten

(24) $$ \begin{equation} \begin{aligned} & {{I}_{mi}}\sum\limits_{j=1}^{i-1}{z_{mi}^{T}{{z}_{qj}}{{{\ddot{q}}}_{j}}}={{I}_{mi}}\sum\nolimits_{j=1}^{i-1}{D_{j}^{i}{{{\ddot{q}}}_{j}}} \\ & =\sum\nolimits_{j=1}^{i-1}{\left[ \begin{matrix} {{I}_{mi}}\hat{D}_{j}^{i} & {{I}_{mi}} \\ \end{matrix} \right]{{\left[ \begin{matrix} {{{\ddot{q}}}_{j}} & \tilde{D}_{j}^{i}{{{\ddot{q}}}_{j}} \\ \end{matrix} \right]}^{T}}} \\ & =\sum\nolimits_{j=1}^{i-1}{U_{j}^{i}} \\ \end{aligned} \end{equation} $$

(25) $$ \begin{equation} \begin{aligned} & {{I}_{mi}}\sum\limits_{j=2}^{i-1}{\sum\limits_{k=1}^{j-1}{z_{mi}^{T}\left( {{z}_{qk}}\times {{z}_{qj}} \right)}{{{\dot{q}}}_{k}}{{{\dot{q}}}_{j}}} \\ & ={{I}_{mi}}\sum\nolimits_{j=2}^{i-1}{\sum\nolimits_{k=1}^{j-1}{\Theta _{kj}^{i}{{{\dot{q}}}_{k}}{{{\dot{q}}}_{j}}}} \\ & =\sum\nolimits_{j=2}^{i-1}{\sum\nolimits_{k=1}^{j-1}{\left[ \begin{matrix} {{I}_{mi}}\hat{\Theta }_{kj}^{i} & {{I}_{mi}} \\ \end{matrix} \right]{{\left[ \begin{matrix} {{{\dot{q}}}_{k}}{{{\dot{q}}}_{j}} & \tilde{\Theta }_{kj}^{i}{{{\dot{q}}}_{k}}{{{\dot{q}}}_{j}} \\ \end{matrix} \right]}^{T}}}} \\ & =\sum\nolimits_{j=2}^{i-1}{\sum\nolimits_{k=1}^{j-1}{V_{kj}^{i}}} \\ \end{aligned} \end{equation} $$

where $$ D_{j}^{i}=z_{mi}^{T}{{z}_{qj}} $$, $$ \hat{D}_{j}^{i}=D_{j}^{i}-\tilde{D}_{j}^{i} $$, $$ \Theta _{kj}^{i}=z_{mi}^{T}\left( {{z}_{qk}}\times {{z}_{qj}} \right) $$, $$ \hat{\Theta }_{kj}^{i}=\Theta _{kj}^{i}-\tilde{\Theta }_{kj}^{i} $$, $$ \hat{D}_{j}^{i} $$ denotes dot product of $$ {{z}_{mi}} $$ and $$ {{z}_{qj}} $$, $$ \tilde{D}_{j}^{i} $$ is alignment error. $$ \hat{\Theta }_{kj}^{i} $$ denotes dot product of unit vector $$ {{z}_{mi}} $$ as well as $$ {{z}_{qk}}\times {{z}_{qj}} $$, $$ \tilde{\Theta }_{kj}^{i} $$ means alignment error.

In addition, $$ {{d}_{i}}({{q}_{i}}) $$ is

(26) $$ \begin{equation} {{d}_{i}}({{q}_{i}})={{d}_{ih}}({{q}_{i}})+{{d}_{is}}({{q}_{i}}) \end{equation} $$

where $$ {{d}_{ih}}({{q}_{i}}) $$ means torque transmission fluctuation at HD, $$ {{d}_{is}}({{q}_{i}}) $$ is measurement disturbance of the sensor.

Reference model Equation (23), IDC are represented as $$ {{I}_{mi}}\sum\limits_{j=1}^{i-1}{z_{mi}^{T}{{z}_{qj}}{{{\ddot{q}}}_{j}}} $$ and $$ {{I}_{mi}}\sum\limits_{j=2}^{i-1}{\sum\limits_{k=1}^{j-1}{z_{mi}^{T}\left( {{z}_{qk}}\times {{z}_{qj}} \right)}{{{\dot{q}}}_{k}}{{{\dot{q}}}_{j}}} $$, while most uncertainties is attributed to joint friction as well as disturbance moment, which exist in terms $$ {{\tilde{F}}_{i}} $$, $$ {{d}_{ih}}({{q}_{i}}) $$ and $$ {{d}_{is}}({{q}_{i}}) $$ respectively. Note that the IDC and model uncertainty is subject to the following attributes.

Property 1. The vector product between $$ {{z}_{mi}} $$$$ {{z}_{qj}} $$$$ {{z}_{qk}} $$ is bounded, i.e. $$ \left| {{D}_{ij}} \right|=\left| z_{mi}^{T}{{z}_{\theta j}} \right|\le 1 $$, $$ \left| {{\Theta }_{ikj}} \right|=\left| z_{mi}^{T}\left( {{z}_{\theta k}}\times {{z}_{\theta j}} \right) \right|\le 1 $$. Meanwhile, when robot joint has been stable, its velocity and acceleration are also bounded, referring to IDC Equations (23) and (24), it can be concluded that if $$ j $$ and $$ k $$ joints are stable, $$ \sum\nolimits_{j=1}^{i-1}{U_{j}^{i}} $$ and $$ \sum\nolimits_{j=2}^{i-1}{\sum\nolimits_{k=1}^{j-1}{V_{kj}^{i}}} $$ are bounded and satisfy relations $$ \left| \sum\nolimits_{j=1}^{i-1}{U_{j}^{i}} \right|\le {{\rho }_{Ui}} $$ and $$ \left| \sum\nolimits_{j=2}^{i-1}{\sum\nolimits_{k=1}^{j-1}{V_{kj}^{i}}} \right|\le {{\rho }_{Vi}} $$, where $$ {{\rho }_{Ui}} $$ and $$ {{\rho }_{Vi}} $$ represent known boundaries.

Remark 1. When joints $$ j $$ and $$ k $$ are stable, terms $$ \sum\nolimits_{j=1}^{i-1}{U_{j}^{i}} $$ and $$ \sum\nolimits_{j=2}^{i-1}{\sum\nolimits_{k=1}^{j-1}{V_{kj}^{i}}} $$ are bounded, which means that the bottom joint $$ i $$-1 is stable when the first joint is controlled. Based on this property, MRM can stabilize joints one by one.

Remark 2. Unlike existing scholars who consider interconnected coupling, including Coriolis force, centrifugal force, and gravity, it is about all robot joints. In this study, because of $$ {{\tau }_{fi}} $$ can reflect the role of the load torque in the space of the $$ i $$th joint subsystem, so that $$ {{Z}_{i}}(q,\dot{q},\ddot{q}) $$ contains only the coupling dynamics part of the bottom joint. This can reduce the amplitude of IDC.

Property 2. Base on Equation (19) and its approximation Equation (20), because $$ {{b}_{fi}} $$$$ {{f}_{ci}} $$$$ {{f}_{si}} $$$$ {{f}_{\tau i}} $$ and their estimates are bounded, $$ {{\tilde{F}}_{i}} $$ is bounded, that is $$ \left| {{{\tilde{F}}}_{i}} \right|\le {{\rho }_{Fil}} $$, where $$ {{\rho }_{Fil}}\text{=}{{\left[ \begin{matrix} {{\rho }_{Fi\text{1}}} & {{\rho }_{Fi\text{2}}} & {{\rho }_{Fi\text{3}}} & {{\rho }_{Fi\text{4}}} \\ \end{matrix} \right]}^{T}} $$ denotes known constant vector. It can be concluded $$ \left| Y({{{\dot{q}}}_{i}}){{{\tilde{F}}}_{i}} \right|\le \left| Y({{{\dot{q}}}_{i}}) \right|{{\rho }_{Fil}} $$.

Property 3. $$ {{f}_{qi}}({{q}_{i}},{{\dot{q}}_{i}}) $$ is bounded $$ \left| {{f}_{qi}}({{q}_{i}},{{{\dot{q}}}_{i}}) \right|\le {{\rho }_{fpi}} $$, where $$ {{\rho }_{fpi}} $$ means known constant which is limited by $$ {{q}_{i}} $$ and speed $$ {{\dot{q}}_{i}} $$.

Property 4. Most of the torque transmission disturbances $$ {{\tau }_{upi}} $$ arise from the elastic compliance of the wave generator and flexspline in the harmonic drive unit. The peak deformation magnitude has been established via factory calibration by the manufacturer. Define $$ {{\tau }_{upi}}={{d}_{ih}}({{q}_{i}}) $$ is a kind of function about the position of robot joints. Therefore, it is easy to get the upper bound of the torque transmission disturbance $$ \left| {{d}_{ih}}({{q}_{i}}) \right|\le {{\rho }_{dhi}} $$.

Property 5. The disturbance $$ {{d}_{is}}({{q}_{i}}) $$ of the sensor is bounded, and its upper bound $$ \left| {{d}_{is}}({{q}_{i}}) \right|\le {{\rho }_{dsi}} $$ is determined by the deviation provided by manufacturer.

Rewriting the dynamic model of $$ i $$th subsystem yields

(27) $$ \begin{equation} {{\ddot{q}}_{i}}=-{{B}_{i}}\left( \begin{aligned} & {{{\hat{b}}}_{fi}}{{{\dot{q}}}_{i}}+\left( {{{\hat{f}}}_{ci}}+{{{\hat{f}}}_{si}}{{e}^{({{{\hat{f}}}_{\tau i}}\dot{q}_{i}^{2})}} \right)sgn ({{{\dot{q}}}_{i}}) \\ & +{{f}_{qi}}({{q}_{i}},{{{\dot{q}}}_{i}})+Y({{{\dot{q}}}_{i}})\tilde{F}\text{+}{{d}_{ih}}({{q}_{i}})+{{d}_{is}}({{q}_{i}}) \\ & +\sum\nolimits_{j=1}^{i-1}{U_{j}^{i}}+\sum\nolimits_{j=2}^{i-1}{\sum\nolimits_{k=1}^{j-1}{V_{kj}^{i}+\frac{{{\tau }_{si}}}{{{\gamma }_{i}}}}} \\ \end{aligned} \right)+{{B}_{i}}{{\tau }_{i}} \end{equation} $$

where $$ {{B}_{i}}={{\left( {{I}_{mi}}{{\gamma }_{i}} \right)}^{-1}}\in {{R}^{+}} $$. Then by defining $$ {{x}_{i}}={{\left[ \begin{matrix} {{x}_{i1}} & {{x}_{i2}} \\ \end{matrix} \right]}^{T}}={{\left[ \begin{matrix} {{q}_{i}} & {{{\dot{q}}}_{i}} \\ \end{matrix} \right]}^{T}} $$, $$ {{u}_{i}}={{\tau }_{i}}\in {{R}^{1\times 1}} $$, state space equation for $$ i $$th joint subsystem is expressed as

(28) $$ \begin{equation} {{S}_{i}}\left\{ \begin{matrix} {{{\dot{x}}}_{i1}}={{x}_{i2}} \\ {{{\dot{x}}}_{i2}}=\phi ({{x}_{i}})+{{h}_{i}}(x)+{{\vartheta }_{i}}({{x}_{i}})+{{B}_{i}}{{u}_{i}} \\ y={{x}_{i1}} \\ \end{matrix} \right. \end{equation} $$

where $$ \phi ({{x}_{i}})\text{=}-{{B}_{i}}\left( {{{\hat{b}}}_{fi}}{{x}_{i2}}+\left( {{{\hat{f}}}_{ci}}+{{{\hat{f}}}_{si}}{{e}^{({{{\hat{f}}}_{\tau i}}x_{i2}^{2})}} \right)sgn ({{x}_{i2}})\text{+}\frac{{{\tau }_{fci}}}{{{\gamma }_{i}}} \right) $$ represents the precise modeling and measurement part of the subsystem dynamics model. The IDC terms $$ {{h}_{i}}(x)=-{{B}_{i}}(\sum\nolimits_{j=1}^{i-1}{U_{j}^{i}}+\sum\nolimits_{j=2}^{i-1}{\sum\nolimits_{k=1}^{j-1}{V_{kj}^{i}}}) $$ and $$ {{\vartheta }_{i}}({{x}_{i}})\text{=}-{{B}_{i}}{{f}_{qi}}({{x}_{i1}},{{x}_{i2}})\text{+}Y({{x}_{i2}})\tilde{F}\text{+}{{d}_{i}}({{x}_{i1}}) $$ are global model uncertainty including friction model error and disturbance.

First, define the overall control torque for each joint subsystem

(29) $$ \begin{equation} {{\tau }_{i}}=\frac{{{\tau }_{fi}}}{{{\gamma }_{i}}}+{{u}_{i}} \end{equation} $$

where $$ {{u}_{i}} $$ denotes the control input of the $$ i $$th joint to be determined. Assuming that the desired trajectory $$ {{q}_{id}} $$ is second-order constrainable and bounded, the following is defined for the controller

(30) $$ \begin{equation} \begin{aligned} & {{e}_{i}}={{q}_{i}}-{{q}_{id}} \\ & {{r}_{i}}={{{\dot{e}}}_{i}}+{{\lambda }_{i}}{{e}_{i}} \\ & {{a}_{i}}={{{\ddot{q}}}_{id}}-2{{\lambda }_{i}}{{{\dot{e}}}_{i}}-{{\lambda }_{i}}^{2}{{e}_{i}} \\ \end{aligned} \end{equation} $$

where $$ {{\lambda }_{i}} $$ is a constant number, refer to the dynamic model of MRM, and compensating each item in the model separately, and define $$ {{u}_{i}} $$ as follows

(31) $$ \begin{equation} {{u}_{i}}={{u}_{ic1}}+{{u}_{ir2}}+{{u}_{in3}}+{{u}_{in4}} \end{equation} $$

where $$ {{u}_{ic1}}\text{=}{{I}_{mi}}{{\gamma }_{i}}{{a}_{i}}+{{\hat{b}}_{i}}{{\dot{q}}_{i}}+\left( {{{\hat{f}}}_{ci}}+{{{\hat{f}}}_{si}}{{e}^{(-{{{\hat{f}}}_{\tau i}}\dot{q}_{i}^{2})}} \right)sgn ({{\dot{q}}_{i}}) $$ is used to compensate for the precise modeling part, including motor moment of inertia and friction force. $$ {{u}_{ir2}} $$ is a robust control term, which is used to compensate for the uncertainty of friction modeling. $$ {{u}_{in\text{3}}} $$ and $$ {{u}_{in\text{4}}} $$ are neural network control terms, which are used to compensate IDC term $$ {{Z}_{i}}(q,\dot{q},\ddot{q}) $$ and disturbance torque term $$ {{d}_{i}}({{q}_{i}}) $$ respectively.

For the uncertainties of friction modeling for the $$ i $$th joint subsystem $$ {{f}_{qi}}({{q}_{i}},{{\dot{q}}_{i}}) $$., if parameter uncertainties are considered as uncertain constants, then uncertainties are compensated by integral compensators. In practice, model parameters are uncertain due to changes in temperature and lubrication. It may not always be constant. Then, by referencing ^[13], combined with the uncertainty compensation of variable parameter model, $$ {{\tilde{F}}_{i}} $$ is decomposed into

(32) $$ \begin{equation} {{\tilde{F}}^{i}}=\tilde{F}_{c}^{i}+\tilde{F}_{v}^{i} \end{equation} $$

where $$ \tilde{F}_{c}^{i} $$ is unknown vector, $$ \tilde{F}_{v}^{i} $$ is a variable and is limited by

(33) $$ \begin{equation} \left| \tilde{F}_{vn}^{i} \right|<\rho _{n}^{i},n = 1,2,3,4 \end{equation} $$

According to the decomposition control design method proposed in the literature^[16], an adaptive compensation control term is designed to compensate constant parameter uncertainty $$ \tilde{F}_{c}^{i} $$, and a robust control term is designed to compensate $$ \tilde{F}_{v}^{i} $$. The designed controller is

(34) $$ \begin{equation} {{u}_{ir2}}\text{=}u_{u}^{i}+Y({{\dot{q}}_{i}})(u_{pc}^{i}+u_{pv}^{i}) \end{equation} $$

where $$ u_{u}^{i} $$ is designed to compensate $$ {{f}_{qi}}({{q}_{i}},{{\dot{q}}_{i}}) $$. Terms $$ u_{pc}^{i} $$ and $$ u_{pv}^{i} $$ are used to compensate the parameter uncertainties $$ \tilde{F}_{c}^{i} $$ and $$ \tilde{F}_{v}^{i} $$. Control terms $$ u_{pc}^{i} $$, $$ u_{pv}^{i} $$ and $$ u_{u}^{i} $$ for the $$ i $$th joint subsystem are defined as:

(35) $$ \begin{equation} u_{u}^{i}=\left\{ \begin{matrix} -{{\rho }_{fi}}\frac{{{r}_{i}}}{\left| {{r}_{i}} \right|}\begin{matrix} {} & \left| {{r}_{i}} \right|>{{\varepsilon }^{i}} \\ \end{matrix} \\ -{{\rho }_{fi}}\frac{{{r}_{i}}}{{{\varepsilon }^{i}}}\begin{matrix} {} & \left| {{r}_{i}} \right|\le {{\varepsilon }^{i}} \\ \end{matrix} \\ \end{matrix} \right. \end{equation} $$

(36) $$ \begin{equation} u_{pc}^{i}=-k\int\limits_{0}^{t}{Y{{({{{\dot{q}}}_{1}})}^{\text{T}}}{{r}_{i}}d\tau } \end{equation} $$

(37) $$ \begin{equation} u_{pvn}^{i}=\left\{ \begin{matrix} -\rho _{n}^{i}\frac{\zeta _{n}^{i}}{\left| \zeta _{n}^{i} \right|}\begin{matrix} {} & \left| \zeta _{n}^{i} \right|>\varepsilon _{pn}^{i} \\ \end{matrix} \\ -\rho _{n}^{i}\frac{\zeta _{n}^{i}}{\varepsilon _{pn}^{i}}\begin{matrix} {} & \left| \zeta _{n}^{i} \right|\le \varepsilon _{pn}^{i} \\ \end{matrix} \\ \end{matrix} \right.,n = 1,2,3,4 \end{equation} $$

where $$ \zeta _{{}}^{i}=Y{{({{\dot{q}}_{1}})}^{\text{T}}}{{r}_{i}} $$, $$ \varepsilon _{{}}^{i},\varepsilon _{pn}^{i} $$ are positive control parameters.

After we completed the compensation design of $$ {{u}_{ic1}} $$ for the precise modeling part and $$ {{u}_{ir2}} $$ for the friction torque respectively. Then we need to design neural network control terms $$ {{u}_{in\text{3}}} $$ and $$ {{u}_{in\text{4}}} $$ to compensate the IDC and disturbance torque terms.

Neural networks are widely recognized for their capacity to approximate arbitrary functions. Their intrinsic self-learning capability eliminates the need for the complex mathematical analyses that are central to conventional adaptive control theory. For highly nonlinear control problems that remain intractable via traditional approaches, the hidden-layer neurons of a multilayer neural network employ activation functions with nonlinear mapping properties, thereby enabling the approximation of any nonlinear function and offering an effective solution to such challenges. Whereas traditional adaptive control methods depend on prior model information - such as a mathematical description of the plant - to design the control scheme, neural-network-based controllers, by virtue of their self-learning ability, require minimal information about the system model or its parameters. As a result, neural network controllers are broadly applicable to control problems involving model uncertainties. Moreover, owing to the massively parallel processing architecture of neural networks, damage to a subset of network nodes does not compromise the overall performance of the entire network, which substantially enhances the fault tolerance of the control system. The radial basis function (RBF) network is structured as a three-layer feedforward architecture that realizes a nonlinear input-output mapping. Since the transformation from the hidden layer to the output layer is linear, the RBF network is recognized as a local function approximator within the broader class of neural networks. Consequently, adopting an RBF network can accelerate learning while circumventing the local minimum problem. A neural network control scheme built upon RBF networks can therefore effectively improve system accuracy, robustness, and adaptability. Because of the IDC $$ {{Z}_{i}}(q,\dot{q},\ddot{q}) $$ and the disturbance torque term $$ {{d}_{i}}({{q}_{i}}) $$ of MRM are highly nonlinear functions, we use RBF neural network to approximate IDC as well as the disturbance torque and compensate them respectively.

According to the Properties 1, 4 and 5, the uncertainty terms $$ {{Z}_{i}}(q,\dot{q},\ddot{q}) $$ and $$ {{d}_{i}}({{q}_{i}}) $$ are bounded. Therefore the uncertainties of modular robot system based on RBF neural network is expressed as follows

(38) $$ \begin{equation} {{Z}_{i}}(q,\dot{q},\ddot{q},W_{zi}^{{}})=W_{zi}^{T}{{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right)+{{e}_{zi}} \end{equation} $$

(39) $$ \begin{equation} {{d}_{i}}\left( {{q}_{i}},{{W}_{di}} \right)=W_{di}^{T}{{\Phi }_{di}}\left( {{q}_{i}} \right)+{{e}_{di}} \end{equation} $$

where $$ W_{zi}^{{}} $$ and $$ W_{di}^{{}} $$ are the ideal neural network weight. Define $$ \hat{W}_{zi}^{{}} $$ and $$ \hat{W}_{di}^{{}} $$ as the estimations of $$ W_{zi}^{{}} $$ and $$ W_{di}^{{}} $$ respectively, $$ {{\tilde{W}}_{di}}={{\hat{W}}_{di}}-{{W}_{di}} $$ and $$ {{\tilde{W}}_{zi}}={{\hat{W}}_{zi}}-{{W}_{zi}} $$ are the estimation errors, $$ {{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right) $$ and $$ {{\Phi }_{di}}\left( {{q}_{i}} \right) $$ are the neural network basis function. $$ {{\hat{d}}_{i}}\left( {{q}_{i}},{{{\hat{W}}}_{di}} \right) $$ is the estimation value of $$ {{d}_{i}}\left( {{q}_{i}},{{W}_{di}} \right) $$, $$ {{\hat{Z}}_{i}}(q,\dot{q},\ddot{q},{{\hat{W}}_{zi}}) $$ is the estimation value of $$ {{Z}_{i}}(q,\dot{q},\ddot{q},{{W}_{zi}}) $$ and they are expressed as

(40) $$ \begin{equation} {{\hat{Z}}_{i}}(q,\dot{q},\ddot{q},{{\hat{W}}_{zi}})=\hat{W}_{zi}^{T}{{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right) \end{equation} $$

(41) $$ \begin{equation} {{\hat{d}}_{i}}\left( {{q}_{i}},{{{\hat{W}}}_{di}} \right)\text{=}\hat{W}_{di}^{T}{{\Phi }_{di}}\left( {{q}_{i}} \right) \end{equation} $$

(42) $$ \begin{equation} {{Z}_{i}}(q,\dot{q},\ddot{q},{{W}_{zi}})-{{\hat{Z}}_{i}}(q,\dot{q},\ddot{q},{{\hat{W}}_{zi}})=\tilde{W}_{zi}^{T}{{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right)+{{e}_{zi}}\text{=}{{e}_{ziH}} \end{equation} $$

(43) $$ \begin{equation} {{d}_{i}}\left( {{q}_{i}},{{W}_{di}} \right)-{{\hat{d}}_{i}}\left( {{q}_{i}},{{{\hat{W}}}_{di}} \right)\text{=}\tilde{W}_{di}^{T}{{\Phi }_{di}}\left( {{q}_{i}} \right)+{{e}_{di}}\text{=}{{e}_{diH}} \end{equation} $$

According to the expression of the neural network in Equations (40) and (41), one has $$ {{u}_{in\text{3}}} $$ and $$ {{u}_{in\text{4}}} $$ to compensate IDC and disturbance torque

(44) $$ \begin{equation} {{u}_{in3}}\text{+}{{u}_{in\text{4}}}\text{=}\hat{W}_{zi}^{T}{{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right)+\hat{W}_{di}^{T}{{\Phi }_{di}}\left( {{q}_{i}} \right) \end{equation} $$

The estimates are updated by

(45) $$ \begin{equation} {{\dot{\hat{W}}}_{zi}}=-k_{1}^{-1}{{e}_{zi}}\Phi (\left| {{r}_{i}} \right|) \end{equation} $$

(46) $$ \begin{equation} {{\dot{\hat{W}}}_{di}}=-k_{2}^{-1}{{e}_{di}}\Phi ({{q}_{i}}) \end{equation} $$

Then, combining with Equations (29), (31), (34) and (44), controller of MRM system is

(47) $$ \begin{equation} \begin{aligned} & {{\tau }_{i}}=\frac{{{\tau }_{fci}}}{{{\gamma }_{i}}}+{{u}_{ic1}}+{{u}_{ir2}}+{{u}_{in3}}+{{u}_{in4}} \\ & =\frac{{{\tau }_{fci}}}{{{\gamma }_{i}}}\text{+}{{I}_{mi}}{{\gamma }_{i}}{{a}_{i}}+{{{\hat{b}}}_{fi}}{{{\dot{q}}}_{i}}+\left( {{{\hat{f}}}_{ci}}+{{{\hat{f}}}_{si}}{{e}^{(-{{{\hat{f}}}_{\tau i}}\dot{q}_{i}^{2})}} \right)sgn ({{{\dot{q}}}_{i}})+u_{u}^{i} \\ & +Y({{{\dot{q}}}_{i}})(u_{pc}^{i}+u_{pv}^{i})+\hat{W}_{zi}^{T}{{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right)+\hat{W}_{di}^{T}{{\Phi }_{di}}\left( {{q}_{i}} \right) \end{aligned} \end{equation} $$

The expression governing the closed-loop behavior of the $$ i $$th joint is written as

(48) $$ \begin{equation} \begin{aligned} & {{M}_{i}}{{{\dot{r}}}_{i}}+\lambda {{M}_{i}}{{r}_{i}}=-{{f}_{qi}}({{q}_{i}},{{{\dot{q}}}_{i}})-Y({{{\dot{q}}}_{i}})({{{\tilde{F}}}_{ci}}+{{{\tilde{F}}}_{vi}})-\frac{{{\tau }_{fci}}}{{{\gamma }_{i}}}+u_{u}^{i}+Y({{{\dot{q}}}_{i}})(u_{pc}^{i}+u_{pv}^{i}) \\ & -{{Z}_{i}}(q,\dot{q},\ddot{q})-{{d}_{i}}({{q}_{i}})+\hat{W}_{zi}^{T}{{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right)+\hat{W}_{di}^{T}{{\Phi }_{di}}\left( {{q}_{i}} \right) \\ \end{aligned} \end{equation} $$

where $$ {{M}_{i}}={{I}_{mi}}{{\gamma }_{i}} $$.

Remark 3. Sign function induces chattering in the system because of dealing with the friction effect. Fortunately, the developed learning-based robust control can solve the chattering phenomenon for favorable tracking performance.

Theorem 1. For an $$ n $$-DOF modular robot manipulator whose joint subsystem dynamic follows Equation (18) and whose model uncertainties are characterized by Equations (24)-(26), application of the control law specified in Equation (47) ensures that the tracking error of each individual joint is ultimately uniformly bounded, while the closed-loop robotic system exhibits asymptotic stability.

Proof: Choosing the Lyapunov candidate as

(49) $$ \begin{equation} {{V}_{i}}=\frac{1}{2}{{M}_{i}}r_{i}^{2}+{{\Theta }_{i}}+\frac{1}{2}{{k}_{\text{1}}}{{\Psi }^{T}}\Psi +\frac{1}{2}{{k}_{\text{2}}}{{\tilde{W}}_{zi}}^{T}{{\tilde{W}}_{zi}}\text{+}\frac{1}{2}{{k}_{\text{3}}}{{\tilde{W}}_{di}}^{T}{{\tilde{W}}_{di}} \end{equation} $$

where $$ \Psi \text{=}\frac{\text{1}}{{{k}_{1}}}{{\tilde{F}}_{c}}-\int\limits_{0}^{t}{Y{{({{{\dot{q}}}_{i}})}^{T}}{{r}_{i}}}d\tau $$, $$ {{k}_{1}},{{k}_{2}},{{k}_{3}} $$ and $$ {{\tilde{F}}_{c}} $$ are constants, $$ \dot{\Psi }=\text{-}Y{{({{\dot{q}}_{i}})}^{T}}{{r}_{i}} $$.

(50) $$ \begin{equation} \begin{aligned} & {{{\dot{V}}}_{i}}={{r}_{i}}\left( \begin{aligned} & -\lambda {{M}_{i}}{{r}_{i}}-{{F}_{qi}}({{q}_{i}},{{{\dot{q}}}_{i}})-Y({{{\dot{q}}}_{1}})({{{\tilde{F}}}_{ci}}+{{{\tilde{F}}}_{vi}}) \\ & +{{u}_{ui}}+Y({{{\dot{q}}}_{1}}){{u}_{pvi}}+Y({{{\dot{q}}}_{1}})(-{{k}_{1}}\int_{0}^{t}{Y{{({{{\dot{q}}}_{1}})}^{T}}{{r}_{i}}}d\tau ) \\ & -{{Z}_{i}}(q,\dot{q},\ddot{q})+\hat{W}_{zi}^{T}{{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right)-{{d}_{i}}({{q}_{i}})+\hat{W}_{di}^{T}{{\Phi }_{di}}\left( {{q}_{i}} \right) \\ \end{aligned} \right) \\ & -{{r}_{i}}(-{{\rho }_{Zi}}+\hat{W}_{zi}^{T}{{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right))sgn ({{r}_{i}})-{{r}_{i}}(-{{\rho }_{Di}}+\hat{W}_{di}^{T}{{\Phi }_{di}}\left( {{q}_{i}} \right))sgn ({{r}_{i}}) \\ & +{{k}_{1}}(\frac{1}{{{k}_{1}}}{{{\tilde{F}}}_{ci}}+\int_{0}^{t}{Y{{({{{\dot{q}}}_{1}})}^{T}}{{r}_{i}}}d\tau )(Y{{({{{\dot{q}}}_{1}})}^{T}}{{r}_{i}})+{{k}_{2}}{{{\tilde{W}}}_{zi}}^{T}{{{\dot{\tilde{W}}}}_{zi}}+{{k}_{3}}{{{\tilde{W}}}_{di}}^{T}{{{\dot{\tilde{W}}}}_{di}} \\ & =-\lambda {{M}_{i}}r_{i}^{2}-{{r}_{i}}{{F}_{qi}}({{q}_{i}},{{{\dot{q}}}_{i}})-{{r}_{i}}Y({{{\dot{q}}}_{1}}){{{\tilde{F}}}_{vi}}+{{r}_{i}}{{u}_{ui}}+{{r}_{i}}Y({{{\dot{q}}}_{1}}){{u}_{pvi}} \\ & +{{r}_{i}}\left( -{{Z}_{i}}(q,\dot{q},\ddot{q})+\hat{W}_{zi}^{T}{{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right)-{{d}_{i}}({{q}_{i}})+\hat{W}_{di}^{T}{{\Phi }_{di}}\left( {{q}_{i}} \right) \right) \\ & -{{r}_{i}}(-{{\rho }_{Zi}}+\hat{W}_{zi}^{T}{{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right))sgn ({{r}_{i}})-{{r}_{i}}(-{{\rho }_{Di}}+\hat{W}_{di}^{T}{{\Phi }_{di}}\left( {{q}_{i}} \right))sgn ({{r}_{i}}) \\ & +{{k}_{2}}{{{\tilde{W}}}_{zi}}^{T}{{{\dot{\tilde{W}}}}_{zi}}+{{k}_{3}}{{{\tilde{W}}}_{di}}^{T}{{{\dot{\tilde{W}}}}_{di}} \\ & =-\lambda {{M}_{i}}r_{i}^{2}+{{r}_{i}}\left( -{{F}_{qi}}({{q}_{i}},{{{\dot{q}}}_{i}})+{{u}_{ui}} \right)+{{r}_{i}}Y({{{\dot{q}}}_{1}})\left( -{{{\tilde{F}}}_{vi}}+{{u}_{pvi}} \right) \\ & -{{r}_{i}}sgn ({{r}_{i}})\left( -{{Z}_{i}}(q,\dot{q},\ddot{q})+{{\rho }_{Zi}}-{{d}_{i}}({{q}_{i}})+{{\rho }_{Di}} \right)+{{k}_{2}}{{{\tilde{W}}}_{zi}}^{T}{{{\dot{\tilde{W}}}}_{zi}}+{{k}_{3}}{{{\tilde{W}}}_{di}}^{T}{{{\dot{\tilde{W}}}}_{di}} \\ \end{aligned} \end{equation} $$

For the $$ i $$th joint, from Equation (37) one can obtain that when $$ \left| {{r}_{i}}Y{{({{{\dot{q}}}_{i}})}^{T}} \right|>{{\varepsilon }_{pni}} $$ there is

(51) $$ \begin{equation} {{r}_{i}}Y{{({{\dot{q}}_{i}})}^{T}}(-{{\tilde{F}}_{vi}}+{{u}_{pvi}})<0 \end{equation} $$

when $$ \left| {{r}_{i}}Y{{({{{\dot{q}}}_{i}})}^{T}} \right|\le {{\varepsilon }_{pni}} $$ there is

(52) $$ \begin{equation} {{r}_{i}}Y{{({{\dot{q}}_{i}})}^{T}}(-{{\tilde{F}}_{vi}}+{{u}_{pvi}})\le {{r}_{i}}Y{{({{\dot{q}}_{i}})}^{T}}\sum\limits_{n=1}^{4}{(\rho _{n}^{i}\frac{{{r}_{i}}Y{{({{{\dot{q}}}_{i}})}^{T}}}{\left| {{r}_{i}}Y{{({{{\dot{q}}}_{i}})}^{T}} \right|}-\rho _{n}^{i}{{\frac{{{r}_{i}}Y({{{\dot{q}}}_{i}})}{{{\varepsilon }_{pni}}}}^{T}})} \end{equation} $$

For the neural network terms there is

(53) $$ \begin{equation} \begin{aligned} & {{k}_{2}}{{{\tilde{W}}}_{zi}}^{T}{{{\dot{\tilde{W}}}}_{zi}}+{{k}_{3}}{{{\tilde{W}}}_{di}}^{T}{{{\dot{\tilde{W}}}}_{di}} \\ & ={{{\tilde{W}}}_{zi}}^{T}({{e}_{ziH}}-\tilde{W}_{zi}^{T}{{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right)){{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right)+{{{\tilde{W}}}_{di}}^{T}({{e}_{diH}}-\tilde{W}_{di}^{T}{{\Phi }_{di}}\left( {{q}_{i}} \right)){{\Phi }_{di}}\left( {{q}_{i}} \right) \\ & ={{e}_{ziH}}\tilde{W}_{zi}^{T}{{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right)-{{\left\| \tilde{W}_{zi}^{T}{{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right) \right\|}^{\text{2}}}+{{e}_{diH}}\tilde{W}_{di}^{T}{{\Phi }_{di}}\left( {{q}_{i}} \right)-{{\left\| \tilde{W}_{di}^{T}{{\Phi }_{di}}\left( {{q}_{i}} \right) \right\|}^{\text{2}}} \\ & \le \frac{1}{2}e_{ziH}^{\text{2}}\text{-}\frac{1}{2}{{\left\| \tilde{W}_{zi}^{T}{{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right) \right\|}^{\text{2}}}+\frac{1}{2}e_{diH}^{\text{2}}\text{-}\frac{1}{2}{{\left\| \tilde{W}_{di}^{T}{{\Phi }_{di}}\left( {{q}_{i}} \right) \right\|}^{\text{2}}} \\ \end{aligned} \end{equation} $$

when $$ {{\tilde{W}}_{zi}} $$ and $$ {{\tilde{W}}_{di}} $$ follow the following constraints the Equation (53) $$ \le 0 $$.

(54) $$ \begin{equation} {{\Omega }_{z}}=\left\{ {{{\tilde{W}}}_{zi}}\left\| {{{\tilde{W}}}_{zi}} \right\|\le \left\| \frac{{{e}_{zM}}}{{{\Phi }_{zM}}} \right\| \right\} \end{equation} $$

(55) $$ \begin{equation} {{\Omega }_{d}}=\left\{ {{{\tilde{W}}}_{di}}\left\| {{{\tilde{W}}}_{di}} \right\|\le \left\| \frac{{{e}_{dM}}}{{{\Phi }_{dM}}} \right\| \right\} \end{equation} $$

From Properties 1,4 and 5, we know that

(56) $$ \begin{equation} -{{r}_{i}}sgn ({{r}_{i}})\left( -{{Z}_{i}}(q,\dot{q},\ddot{q})+{{\rho }_{Zi}}-{{d}_{i}}({{q}_{i}})+{{\rho }_{Di}} \right)\le 0 \end{equation} $$

Because the term Equation (52) reaches its maximum at $$ \left| {{r}_{i}}Y{{({{{\dot{q}}}_{i}})}^{T}} \right|\le \frac{{{\varepsilon }_{pni}}}{2} $$, combined with Equations (53) and (56) obtain that

(57) $$ \begin{equation} \begin{aligned} & {{{\dot{V}}}_{i}}\le \sum\limits_{n=1}^{4}{(\frac{\rho _{n}^{i}{{\varepsilon }_{pni}}}{4})+\frac{\rho _{fpi}^{{}}{{\varepsilon }_{i}}}{4}}-\lambda {{M}_{i}}r_{i}^{2}+\frac{1}{2}e_{ziH}^{\text{2}} \\ & -\frac{1}{2}{{\left\| \tilde{W}_{zi}^{T}{{\Phi }_{zi}}\left( \left| {{r}_{i}} \right| \right) \right\|}^{\text{2}}}+\frac{1}{2}e_{diH}^{\text{2}}\text{-}\frac{1}{2}{{\left\| \tilde{W}_{di}^{T}{{\Phi }_{di}}\left( {{q}_{i}} \right) \right\|}^{\text{2}}} \\ & -{{r}_{i}}sgn ({{r}_{i}})\left( -{{Z}_{i}}(q,\dot{q},\ddot{q})+{{\rho }_{Zi}}-{{d}_{i}}({{q}_{i}})+{{\rho }_{Di}} \right) \\ \end{aligned} \end{equation} $$

According to Equation (57), we can know that Lyapunov functions can only be found if the following relations are satisfied

(58) $$ \begin{equation} \left| {{r}_{i}} \right|\ge \sqrt[3]{\frac{\sum\limits_{n=1}^{4}{(\rho _{n}^{i}{{\varepsilon }_{pni}})+\rho _{fpi}^{{}}{{\varepsilon }_{i}}}}{4\lambda {{M}_{i}}}} \end{equation} $$

Define

(59) $$ \begin{equation} {{r}_{ir}}=\left\{ {{r}_{i}}\in R|r_{i}^{2}\le \frac{\sum\limits_{n=1}^{4}{(\rho _{n}^{i}{{\varepsilon }_{pni}})+\rho _{fpi}^{{}}{{\varepsilon }_{i}}}}{2\lambda {{M}_{i}}} \right\} \end{equation} $$

Then, on the surface of $$ {{r}_{ir}} $$ and $$ \partial {{r}_{ir}} $$, obtain that

(60) $$ \begin{equation} {{\dot{V}}_{i}}\le -\frac{\sum\limits_{n=1}^{4}{(\rho _{n}^{i}{{\varepsilon }_{pni}})+\rho _{fpi}^{{}}{{\varepsilon }_{i}}}}{4} \end{equation} $$

Denote $$ {{T}_{ir}} $$ as the time required for the solution trajectory to reach the surface $$ \partial {{r}_{ir}} $$

(61) $$ \begin{equation} {{V}_{i}}({{r}_{ir}}({{T}_{ir}}))-{{V}_{i}}({{r}_{ir}}(0))\le -\frac{\sum\limits_{n=1}^{4}{(\rho _{n}^{i}{{\varepsilon }_{pni}})+\rho _{fpi}^{{}}{{\varepsilon }_{i}}}}{4}{{T}_{ir}} \end{equation} $$

(62) $$ \begin{equation} {{T}_{ir}}\le \frac{4({{V}_{i}}({{r}_{ir}}({{T}_{ir}}))-{{V}_{i}}({{r}_{ir}}(0)))}{\sum\limits_{n=1}^{4}{(\rho _{n}^{i}{{\varepsilon }_{pni}})+\rho _{fpi}^{{}}{{\varepsilon }_{i}}}} \end{equation} $$

According to Equation (30), the boundedness of $$ {{r}_{i}} $$ means boundedness of $$ {{e}_{i}} $$ as well as $$ {{\dot{e}}_{i}} $$, so that theorem is proved.

Remark 4. According to the Lyapunov stability proof, it can be seen that the system can achieve stability as time approaches infinity. Finite-time and exponential stability will be the focus of our next research work. Additionally, through the experimental verification in the next section, it can be observed that the system can maintain stability within a very short period of time, ensuring good real-time performance.

Figures 2 and 3 present the trajectory tracking curves of the modular robot. This part of the experiment compares the trajectory-tracking performance of the conventional robust control method ^[26-28] with that of the proposed robust neural network control method. The experimental curves indicate that, while both control methods are capable of tracking the desired trajectory reasonably well, the conventional robust control method still exhibits relatively large tracking errors at certain points along the trajectory. In contrast, the robust neural network-based control method achieves superior trajectory-tracking performance, owing to its ability to more accurately approximate the MRM model and compensate for model uncertainties.

Figures 4 and 5 present the trajectory tracking error curves of the modular robot. This segment of the experiment compares the tracking errors obtained using the conventional robust control method with those achieved by the robust neural network control method proposed in this paper. The experimental curves clearly indicate that a pronounced chattering effect is evident in the error profile when the conventional robust control method is employed. By contrast, owing to the neural network's capacity to approximate and compensate for model uncertainties and coupling terms, the proposed robust neural network control method accomplishes accurate compensation of model uncertainties, thereby effectively suppressing joint chattering, reducing control error, and yielding superior control accuracy. As a result, the trajectory tracking performance of both joints is substantially enhanced.

Figures 6 and 7 present the motor output torque curves of the modular robot. This part of the experiment compares the motor output torque obtained using the traditional robust control method with that obtained using the robust neural network control method proposed in this paper. It can be observed from the experimental curves that the motor output torque under the traditional robust control method exhibits a pronounced chattering effect, whereas the motor output torque under the proposed robust neural network control method is relatively smooth. The control torque root mean square of the joint 1’s existed method is 0.18, and the developed method is 0.12. The control torque root mean square of the joint 2’s existed method is 0.2, and the developed method is 0.12. The value of root mean square is suppressed by about 35%. The calculate method is as follows: $$ \left( 0.18-0.12 \right)/0.18\approx 33.3\%, \left( 0.2-0.12 \right)/0.2\approx 40\%. $$

Figures 8 and 9 depict the weight variation curves of the neural network for joint 1 and joint 2 of the modular robot, respectively. In this experiment, five neural nodes were configured to approximate the model uncertainties of the robot. As can be observed from the experimental curves, the neural network weights oscillate in a regular manner within a bounded range.

Made substantial contributions to conception and design of the study and performed data analysis and interpretation: Ji, Z.

Performed data acquisition, as well as provided administrative, technical, and material support: An, T.

The data presented in this study are available on request from the corresponding author because the experimental data were generated by the experimental platform and cannot be used independently.

Not applicable.

The work is supported by the Scientific Technological Development Plan Project in Jilin Province of China (20260602029RC).

All authors declared that there are no conflicts of interest.

Not applicable.

Not applicable.

The Author(s) 2026.