Barrier Function-Based Integral Sliding Mode Controller Design for a Single-Link Rotary Flexible Joint Robot

As seen in Fig. 1, it consists of double rotary arms mounted on a base platform – a 29.8 cm extended arm attached with springs holding a short arm. The short arm’s placement on the long arm determines the flexible joint’s vibration effects. The arm’s flexibility comes from the spring stiffness. The base is fixed on a Quanser SRV02 rotary servo motor, which has a DC motor enclosed in an aluminum frame connected to a gearbox. This complete servo system contains gears, a potentiometer, a tachometer, encoders, and a DC motor. A tachometer measures the rotary arm’s angular displacement, while a potentiometer measures the load gear orientation to obtain angular velocity. Two encoders measure joint rotation and servo load shaft angle.

Figure 1:

In Quanser’s study [36], the mathematical model of Quanser’s rotary single-link FJR is presented. The schematic of the rotary flexible joint arm module is shown in Fig. 2, where it operates in a horizontal plane; thus, gravity terms are not considered. Consequently, θ and α represent the angular position of the motor and the vibration/deflection angle of the link (flexible joint), respectively. Euler–Lagrange’s method is often utilized for more complex systems, such as a robot manipulator with multiple joints. The dynamics of the FJR can be modeled using differential equations representing the link and motor dynamics. The dynamical model of the system is presented as follows [31]: (1) Jeqθ¨+Jarm(θ¨+a¨)=τ−Beqθ¨ {J_{eq}}\ddot \theta + {J_{arm}}\left( {\ddot \theta + \ddot a} \right) = \tau - {B_{eq}}\ddot \theta (2) Jarm(θ¨+a¨)+Ksα−d(t)=0 {J_{arm}}\left( {\ddot \theta + \ddot a} \right) + {K_s}\alpha - d\left( t \right) = 0

Figure 2:

In this dynamic model, there are two kinds of inertia. The equivalent moment of inertia of the rotary arm is expressed by J_eq, whereas the inertia of the link is expressed by J_arm. Further, B_eq and K_s represent the viscous friction coefficient of the servo and linear spring stiffness, respectively, and d(t) is the external disturbance acting on the unactuated dynamics.

The torque (τ) that is enforced at the base of the rotary manipulator arm is generated while the input voltage is applied, which is define as: (3) τ=KtKgηmηg(Vm−KgKmθ˙)Rm \tau = {{{K_t}{K_g}{\eta _m}{\eta _g}\left( {{V_m} - {K_g}{K_m}\dot \theta } \right)} \over {{{\rm{R}}_m}}}

All the symbols in the above equation are defined, and their numerical values are presented in Table 1 below [20].

Hence, by substituting Eq. (3) into Eq. (1), Eqs. (4) and (5) can be written as: (4) α¨=−θ¨−KsJarmα−d(t)Jarm \ddot \alpha = - \ddot \theta - {{{K_s}} \over {{J_{arm}}}}\alpha - {{d\left( t \right)} \over {{J_{arm}}}} (5) θ¨=KsJeqα−F1θ˙+F2Vm \ddot \theta = {{{K_s}} \over {{J_{eq}}}}\alpha - {F_1}\dot \theta + {F_2}{V_m} where F1=BeqRm+ηmηgKtKmKg2JeqRmF2=ηmηgKtKgJeqRm \matrix{ {{F_1} = {{{B_{eq}}{{\rm{R}}_m} + {\eta _m}{\eta _g}{K_t}{K_m}K_g^2} \over {{J_{eq}}{{\rm{R}}_m}}}} \hfill \cr {{F_2} = {{{\eta _m}{\eta _g}{K_t}{K_g}} \over {{J_{eq}}{{\rm{R}}_m}}}} \hfill \cr } V_m is the motor control voltage; the target is to design a controller to track y = α + θ its desired trajectory. Therefore, Eqs. (4) and (5) are rewritten as (6) y¨=−KsJarm(y−θ)−d(t)Jarm {\ddot y = {{ - {K_s}} \over {{J_{arm}}}}\left( {y - \theta } \right) - {{d\left( t \right)} \over {{J_{arm}}}}} (7) θ¨=KsJeq(y−θ)−F1θ˙+F2Vm {\ddot \theta = {{{K_s}} \over {{J_{eq}}}}\left( {y - \theta } \right) - {F_1}\dot \theta + {F_2}{V_m}}

For the convenience of controller design, the following state vector is defined: x1=y,x2=y˙,x3=θ,x4=θ˙,y=x1 {x_1} = y,{x_2} = \dot y,{x_3} = \theta ,{x_4} = \dot \theta ,y = {x_1}

So, the dynamics of Eqs. (6) and (7) can be written in state space form as (8) x˙1=x2x˙2=a1(x1−x3)+TLx˙3=x4x˙4=b1(x1−x3)−F1x4+F2Vm \matrix{ {{{\dot x}_1} = {x_2}} \hfill \cr {{{\dot x}_2} = {a_1}\left( {{x_1} - {x_3}} \right) + {T_L}} \hfill \cr {{{\dot x}_3} = {x_4}} \hfill \cr {{{\dot x}_4} = {b_1}\left( {{x_1} - {x_3}} \right) - {F_1}{x_4} + {F_2}{V_m}} \hfill \cr } where a1=−KsJarm,b1=KsJeq, and TL=−d(t)Jarm. {a_1} = {{ - {K_s}} \over {{J_{arm}}}},{b_1} = {{{K_s}} \over {{J_{eq}}}},\;{\rm{and}}\;{T_L} = - {{d\left( t \right)} \over {{J_{arm}}}}.

The uncertainty in the above equations consists of the joint stiffness value, i.e., K_s, which can be obtained as: Ks=Ksn+ΔKs then, a1=a1n+Δa1, and b1=b1n+Δb1 {K_s} = {K_{sn}} + \Delta {K_s}\;{\rm{then,}}\;{a_1} = {a_{1n}} + \Delta {a_1},\;{\rm{and}}\;{b_1} = {b_{1n}} + \Delta {b_1}

Also, uncertainties in the system model coefficients, F₁ and F₂, can be obtained as: F1=F1n+ΔF1,F2=F2n+ΔF2 {F_1} = {F_{1n}} + \Delta {F_1},{F_2} = {F_{2n}} + \Delta {F_2}

Thereby, the system dynamics of FJR, as it is the basis for the designed controller in the next section, accordingly becomes: (9) x˙1=x2x˙2=a1n(x1−x3)+δ1x˙3=x4x˙4=b1n(x1−x3)−F1nx4+F2nVm+δ2 \matrix{ {{{\dot x}_1} = {x_2}} \hfill \cr {{{\dot x}_2} = {a_{1n}}\left( {{x_1} - {x_3}} \right) + {\delta _1}} \hfill \cr {{{\dot x}_3} = {x_4}} \hfill \cr {{{\dot x}_4} = {b_{1n}}\left( {{x_1} - {x_3}} \right) - {F_{1n}}{x_4} + {F_{2n}}{V_m} + {\delta _2}} \hfill \cr } where δ₁ and δ₂ are the unactuated and actuated perturbations, respectively, and can be written as: (10) δ1=Δa1(x1−x3)+TLδ2=Δb1(x1−x3)−ΔF1x4+ΔF2Vm \matrix{ {{\delta _1} = \Delta {a_1}\left( {{x_1} - {x_3}} \right) + {T_L}} \hfill \cr {{\delta _2} = \Delta {b_1}\left( {{x_1} - {x_3}} \right) - \Delta {F_1}{x_4} + \Delta {F_2}{V_m}} \hfill \cr }

Definition [39]: Let us assume that some ε > 0 is given and fixed; a BF is defined as an even continuous function f: z ∈ [−ε, ε] → g(z) ∈ [b, ∞] strictly increasing on [0, ε].

lim_|z|→ε g(z) = ∞₊

In this paper, two different classes of BFs are considered: (a)

Positive-definite BFs (PBFs): gp(z)=εFε−|z|,gp(0)=F {g_p}\left( z \right) = {{\varepsilon F} \over {\varepsilon - \left| z \right|}},{g_p}\left( 0 \right) = F .

The PBFs g_ps(z) in (b) were chosen and will be used when simulating the FJR system in this paper.

The problem with the robustness of the conventional SMC is that it is not guaranteed during the reaching phase, although the parametric variations and external disturbances during the sliding mode alone can be achieved. However, this problem is addressed via the proposed controller in this research, which is ISMC based on the barrier function approach previously proposed in the researches of Abd and Al-Samarraie [38] and Mohammadridha and Al-Samarraie [40]. This controller was able to eliminate the reaching phase and all parameter uncertainties from the first instant to ensure robustness from the beginning of a process.

As mentioned above, the ISMC is designed for each subsystem; so, the unactuated subsystem, according to Eq. (9), is (11) x˙1=x2x˙2=a1n(x1−x3)+δ1 \matrix{ {{{\dot x}_1} = {x_2}} \hfill \cr {{{\dot x}_2} = {a_{1n}}\left( {{x_1} - {x_3}} \right) + {\delta _1}} \hfill \cr }

Let us define the following state transformation. (12) vc=(x1−x3) {v_c} = \left( {{x_1} - {x_3}} \right)

Accordingly, the mathematical model in Eq. (11) becomes as follows: (13) x˙1=x2x˙2=a1nvc+δ1 \matrix{ {{{\dot x}_1} = {x_2}} \hfill \cr {{{\dot x}_2} = {a_{1n}}{v_c} + {\delta _1}} \hfill \cr } where v_c is a new state that can be defined as a controller.

By introducing the new state, the back-stepping approach can be applied to the system dynamics in Eq. (14), which is regarded as a virtual control, as seen later in this section.

Now, let us consider the unactuated subsystem with v_c = u, where u is the virtual controller. (14) x˙1=x2x˙2=a1nu+δ1 \matrix{ {{{\dot x}_1} = {x_2}} \hfill \cr {{{\dot x}_2} = {a_{1n}}u + {\delta _1}} \hfill \cr }

The error functions e₁ and e₂ are formally defined, respectively, as the quantitative difference or discrepancy between the intended/target input (i.e., reference value) and the real/actual measured values of the output. This definition is demonstrated through the mathematical Eq. (15) below. (15) e1=x1−xde2=x2−x˙d \matrix{ {{e_1} = {x_1} - {x_d}} \hfill \cr {{e_2} = {x_2} - {{\dot x}_d}} \hfill \cr } where x_d is the desired (or reference) signal; hence, Eq. (16) in terms of the error functions is given by (16) e˙1=e2e˙2=a1nu+δ1−x¨d \matrix{ {{{\dot e}_1} = {e_2}} \hfill \cr {{{\dot e}_2} = {a_{1n}}u + {\delta _1} - {{\ddot x}_d}} \hfill \cr }

In this work, the proposed virtual controller u for the unactuated subsystem in Eq. (14) is the ISMC based on the barrier function, which is given by (17) u=1a1n(uo+us) u = {1 \over {{a_{1n}}}}\left( {{u_o} + {u_s}} \right) where u_o is the nominal control, which will be determined later, while u_s is PBFs. For the unactuated subsystem, the sliding variable s₁ is defined as (18) s1=e2+z1. {s_1} = {e_2} + {z_1}.

It consists of two main parts: e₂ is the derivative of the tracking error and z₁ is the integral term, with s₁, z₁ ∈ R. Then, the integral sliding manifold derivative is (19) s˙1=e˙2+z˙1. {\dot s_1} = {\dot e_2} + {\dot z_1}.

By substituting Eqs. (16) and (17) in Eq. (19), the following equation can be obtained: (20) s˙1=uo+us+δ1−x¨d+z¨1 {\dot s_1} = {u_o} + {u_s} + {\delta _1} - {\ddot x_d} + {\ddot z_1} where x¨d {\ddot x_d} , is the second derivative of the desired input.

Now, the derivative of the integral part is defined as (21) z˙1=x¨d−uo {\dot z_1} = {\ddot x_d} - {u_o}

Accordingly, s˙1 {\dot s_1} becomes (22) s˙1=us+δ1. {\dot s_1} = {u_s} + {\delta _1}.

The equivalent system dynamics [41], which represent the system dynamics during the sliding mode, can be obtained from the system dynamics described in Eq. (23) by substituting u_s = −δ₁. So, following the substitution of u and u_s in Eq. (17), the system dynamic from Eq. (16) would assume the following form: (23) e˙1=e2e˙2=uo−x¨d \matrix{ {{{\dot e}_1} = {e_2}} \hfill \cr {{{\dot e}_2} = {u_o} - {{\ddot x}_d}} \hfill \cr }

Note that the unactuated subsystem is equivalent to the subsystem given in Eq. (24); just after v_c, it becomes equal to the virtual controller u, which is the task of the actual controller. Therefore, a nominal controller u_o can be selected as in Eq. (24), which makes the origin of the error dynamics in Eq. (24) globally asymptotically stable. (24) uo=x¨d−c1e1−c2e2 {u_o} = {\ddot x_d} - {c_1}{e_1} - {c_2}{e_2} where c₁ and c₂ are positive constant values chosen based on the desired characteristic. Consequently, the proposed ISMC law based on using barrier function is a continuous control law, where u_s can take the following form: (25) us=−λ1s1ε1−|s1| {u_s} = {{ - {\lambda _1}{s_1}} \over {{\varepsilon _1} - \left| {{s_1}} \right|}} where ε₁ > 0 is the thickness of the boundary layer, while λ₁ > 0 represents the control gain, which is not required to be computed as the gain in the classical ISMC because the bond of perturbation must be known. So, only a suitable value for ε₁ according to the wanted accuracy has to be selected.

Thereby, the chattering and reaching phase will be eliminated from the beginning of the process when selecting ε₁ as small as possible. Furthermore, the steady-state error is a function to ε₁, and so it becomes smaller for smaller ε₁. To this end, the actuated subsystem is given according to Eq. (9): (26) x˙3=x4x˙4=b1n(x1−x3)−F1nx4+F2nVm+δ2 \matrix{ {{{\dot x}_3} = {x_4}} \hfill \cr {{{\dot x}_4} = {b_{1n}}\left( {{x_1} - {x_3}} \right) - {F_{1n}}{x_4} + {F_{2n}}{V_m} + {\delta _2}} \hfill \cr }

The controller’s task V_m in the actuated subsystem is to make the horizontal link position θ track the desired angle while reducing the vibration angle α of the rotary FJR. Similarly, the following steps are executed to design the actual controller by considering the state transformation v_c dynamic in Eq. (12). The error functions e₃ and e₄ are defined as (27) e3=vc−ue4=v˙c−u˙ \matrix{ {{e_3} = {v_c} - u} \hfill \cr {{e_4} = {{\dot v}_c} - \dot u} \hfill \cr } where u is the desired input (virtual controller) for the unactuated subsystem; accordingly, Eq. (27) in terms of the error functions (Eq. (28)) is provided by (28) e˙3=e4e˙4=Fn+δ1−F2nVm−δ2−u¨ \matrix{ {{{\dot e}_3} = {e_4}} \hfill \cr {{{\dot e}_4} = {F_n} + {\delta _1} - {F_{2n}}{V_m} - {\delta _2} - \ddot u} \hfill \cr } where (29) Fn=a1n(x1−x3)−b1n(x1−x3)+F1nx4 {F_n} = {a_{1n}}\left( {{x_1} - {x_3}} \right) - {b_{1n}}\left( {{x_1} - {x_3}} \right) + {F_{1n}}{x_4}

Thus, the proposed controller for the actuated subsystem in Eq. (26) is the ISMC based on the barrier function. Therefore, the V_m controller is given as follows: (30) Vm=−1F2n(Vmn+Vms) {V_m} = {{ - 1} \over {{F_{2n}}}}\left( {{V_{mn}} + {V_{ms}}} \right) where V_mn is the nominal control and V_ms is PBFs. For the actuated subsystem, the sliding variable s₂ is defined as: (31) s2=e4+z2 {s_2} = {e_4} + {z_2}

It consists of two main parts: e₄ is the derivative of the tracking error and z₂ is the integral term with s₂, z₂ ∈ R. Then, the integral sliding manifold derivative is (32) s˙2=e˙4+z˙2 {\dot s_2} = {\dot e_4} + {\dot z_2}

By substituting Eqs. (28) and (30) into Eq. (32), the following is obtained: (33) s˙2=Fn+ +Vmn+Vms−δ˜+z˙2 {\dot s_2} = {F_n} + \; + {V_{mn}} + {V_{ms}} - \tilde \delta + {\dot z_2} where δ˜=δ1−δ2−u¨ \tilde \delta = {\delta _1} - {\delta _2} - \ddot u .

Now, let the derivative of the integral part be defined as (34) z˙2=−Fn−Vmn {\dot z_2} = - {F_n} - {V_{mn}}

Accordingly, s˙2 {\dot s_2} becomes (35) s˙2=δ˜+Vms {\dot s_2} = \tilde \delta + {V_{ms}}

Then, in the equivalent mode, Vms=−δ˜ {V_{ms}} = - \tilde \delta . Therefore, from Eq. (28), we obtain the following equivalent dynamics: (36) e˙3=e4e˙4=Fn+Vmn \matrix{ {{{\dot e}_3} = {e_4}} \hfill \cr {{{\dot e}_4} = {F_n} + {V_{mn}}} \hfill \cr } where the nominal control V_mn is selected as: (37) Vmn=−Fn+v {V_{mn}} = - {F_n} + v where v is the equivalent controller.

Consequently, the proposed ISMC law based on using a barrier function for the actuated subsystem is also a continuous control law where V_ms can take the following form: (38) Vms=−λ2s2ε2−|s2| {V_{ms}} = {{ - {\lambda _2}{s_2}} \over {{\varepsilon _2} - \left| {{s_2}} \right|}} where ε₂ > 0 is the thickness of the boundary layer, while λ₂ > 0 represents the control gain. Thereby, only a suitable value of ε₂ needs to be selected according to the desired accuracy so that the chattering and reaching phases are eliminated from the initiation of the process due to selecting ε₂ as small as possible. Therefore, the steady-state error is a function of ε₂, and so it becomes smaller for smaller ε₂.

The nominal control part of the motor voltage is designed based on the design of V_mn above. As a first step, the dynamic of FJR in terms of the error functions is introduced in the following way, that is to say from Eqs. (17) and (27), we have: vc=e3+u=e3+1a1n(uo+us) {v_c} = {e_3} + u = {e_3} + {1 \over {{a_{1n}}}}\left( {{u_o} + {u_s}} \right)

Then Eq. (16) becomes: (39) e˙1=e2e˙2=uo+us+a1ne3+δ^1 \matrix{ {{{\dot e}_1} = {e_2}} \hfill \cr {{{\dot e}_2} = {u_o} + {u_s} + {a_{1n}}{e_3} + {{\hat \delta }_1}} \hfill \cr }

Eq. (39) is the unactuated subsystem in terms of the error functions. In the next step, the actuated subsystem in Eqs. (28) and (30) are rewritten in the following form: (40) e˙3=e4e˙4=Fn+Vmn+Vms+δ˜ \matrix{ {{{\dot e}_3} = {e_4}} \hfill \cr {{{\dot e}_4} = {F_n} + {V_{mn}} + {V_{ms}} + \tilde \delta } \hfill \cr }

And from Eq. (37), V_mn = −F_n + v, the dynamics of FJR in terms of the error functions is outlined using the following mathematical expression: (41) [e˙1e˙2e˙3e˙4]=[010000a1n000010000][e1e2e3e4]+[0001]v+[0us+δ^10Vms+δ˜] \left[ {\matrix{ {{{\dot e}_1}} \cr {{{\dot e}_2}} \cr {{{\dot e}_3}} \cr {{{\dot e}_4}} \cr } } \right] = \left[ {\matrix{ 0 & 1 & 0 & 0 \cr 0 & 0 & {{a_{1n}}} & 0 \cr 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 0 \cr } } \right]\left[ {\matrix{ {{e_1}} \cr {{e_2}} \cr {{e_3}} \cr {{e_4}} \cr } } \right] + \left[ {\matrix{ 0 \cr 0 \cr 0 \cr 1 \cr } } \right]v + \left[ {\matrix{ 0 \cr {{u_s} + {{\hat \delta }_1}} \cr 0 \cr {{V_{ms}} + \tilde \delta } \cr } } \right]

Since the sliding mode is introduced in the ISMC design from the first instant, one can use the equivalent concept to design v by writing the error model, which is given in Eq. (42) in the equivalent form, i.e., (42) [e˙1e˙2e˙3e˙4]=[010000a1n000010000][e1e2e3e4]+[0001]v+[0us+δ^10Vms+δ˜]equivalent \left[ {\matrix{ {{{\dot e}_1}} \cr {{{\dot e}_2}} \cr {{{\dot e}_3}} \cr {{{\dot e}_4}} \cr } } \right] = \left[ {\matrix{ 0 & 1 & 0 & 0 \cr 0 & 0 & {{a_{1n}}} & 0 \cr 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 0 \cr } } \right]\left[ {\matrix{ {{e_1}} \cr {{e_2}} \cr {{e_3}} \cr {{e_4}} \cr } } \right] + \left[ {\matrix{ 0 \cr 0 \cr 0 \cr 1 \cr } } \right]v + {\left[ {\matrix{ 0 \cr {{u_s} + {{\hat \delta }_1}} \cr 0 \cr {{V_{ms}} + \tilde \delta } \cr } } \right]_{{\rm{equivalent}}}}

But [0us+δ^10Vms+δ˜]equivalent=[0[us]equivalent+δ^10[Vms]equivalent+δ˜]=[0000] {\left[ {\matrix{ 0 \cr {{u_s} + {{\hat \delta }_1}} \cr 0 \cr {{V_{ms}} + \tilde \delta } \cr } } \right]_{{\rm{equivalent}}}} = \left[ {\matrix{ 0 \cr {{{\left[ {{u_s}} \right]}_{{\rm{equivalent}}}} + {{\hat \delta }_1}} \cr 0 \cr {{{\left[ {{V_{ms}}} \right]}_{{\rm{equivalent}}}} + \tilde \delta } \cr } } \right] = \left[ {\matrix{ 0 \cr 0 \cr 0 \cr 0 \cr } } \right] where [us]equivalent=−δ^1 {\left[ {{u_s}} \right]_{{\rm{equivalent}}}} = - {\hat \delta _1} and [Vms]equivalent=−δ˜ {\left[ {{V_{ms}}} \right]_{{\rm{equivalent}}}} = - \tilde \delta , resulting in the dynamics of FJR in the equivalent form from the first instant given by: (43) [e˙1e˙2e˙3e˙4]=[010000a1n000010000][e1e2e3e4]+[0001]=Ae+Bv \left[ {\matrix{ {{{\dot e}_1}} \cr {{{\dot e}_2}} \cr {{{\dot e}_3}} \cr {{{\dot e}_4}} \cr } } \right] = \left[ {\matrix{ 0 & 1 & 0 & 0 \cr 0 & 0 & {{a_{1n}}} & 0 \cr 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 0 \cr } } \right]\left[ {\matrix{ {{e_1}} \cr {{e_2}} \cr {{e_3}} \cr {{e_4}} \cr } } \right] + \left[ {\matrix{ 0 \cr 0 \cr 0 \cr 1 \cr } } \right] = Ae + Bv

One can easily show that A and B are controllable pairs by using linear state feedback for v. (44) v=−Ke v = - Ke where K = [c₃ c₄ c₅ c₆]^T. The matrix A − BK is Hurwitz with the desired characteristics. Accordingly, the nominal voltage control signal becomes (45) Vmn=−Fn−Ke {V_{mn}} = - {F_n} - Ke which will render the origin e = 0 asymptotically stable.

In this section, a sliding mode observer (SMO) is proposed [42]. The first derivative of the virtual controller u(u˙) u\left( {\dot u} \right) is required in the error function to design the controller of the actuated subsystem. So, the SMD is used to estimate the first derivatives of u. The SMD is constructed as in Eq. (46): (46) so=u−ηη˙=ko*tan−1(γoso)v˙o=1τo(−vo+ko*tan−1(γoso)) \matrix{ {{s_o} = u - \eta } \hfill \cr {\dot \eta = {k_o}\,^*\,{{tan }^{ - 1}}\left( {{\gamma _o}{s_o}} \right)} \hfill \cr {{{\dot v}_o} = {1 \over {{\tau _o}}}\left( { - {v_o} + {k_o}\,^*\,{{tan }^{ - 1}}\left( {{\gamma _o}{s_o}} \right)} \right)} \hfill \cr } where s_o is the SMD variable, and k_o and γ_o are the differentiator parameters. The second line in Eq. (46) is the SMD dynamics, while the third line is the low-pass filter with time constants τ_o. Thus, the low-pass filter outputs the estimation of the first derivatives for u.

Remark: In Eq. (46), the subscript (o) refers to the observer.

In this case, the desired position of the link of FJR is sinusoidal input at x_d = 35sin (2π0.1t) deg. Figs. 3 and 4 show the sliding variable response for the virtual (unactuated) and actual (actuated) ISMC-based barrier functions. It can be seen that the sliding variable of the two controllers is smooth and does not suffer from the chattering phenomenon. As the figures demonstrate, the controllers work to regulate the sliding variable to zero levels after less than 0.15 s, and do not exceed the predefined neighborhood around zero (ε₁, ε₂). The suggested controller successfully removes the reaching phase and disturbances, ensuring robustness right from the start of a process. This highlights that the FJR system, operating under the ISMC-based barrier function, accomplishes effective stabilization. The position link of the FJR system and the vibration angle are shown in Fig. 5, where the tracking performance of the proposed controller was demonstrated when the desired trajectory to be tracked is a sinusoidal signal. It is obvious that the proposed controller needs less than 0.1 s to make the system follow the input waveform and be very close to it after that. Another note from Fig. 5 in the zoomed view is that during the time interval between 0 s and 0.2 s, there is a small increase in the amplitude of vibration angle (α), but it is satisfactory, and then the rest of this angle goes to less than (thereafter, as illustrated in Fig. 6. The position tracking error e₁ = x₁ − x_d is plotted for this scenario, and the maximum position tracking error does not overtake 0.1, as shown in Fig. 7.

Figure 3:

Figure 4:

Figure 5:

Figure 6:

Figure 7:

Additionally, the control effort represented by V_m was required for the tracking process, as clarified in Fig. 8. It is obvious in the zoomed version of this figure that the proposed controller is smooth due to using the barrier function with ISMC instead of the signum function because the ISMC-based barrier function does not produce chattering in the invariant neighborhood of s, which is the undesirable behavior in SMC. Therefore, it does not need continuous approximation (arctan) or a low-pass filter.

Figure 8:

To investigate the ability and effectiveness of the designed controller, a square-wave profile with an amplitude of ±35° is used as another desired trajectory that the arm of FJR should track. Figs. 9 and 10 below depict the tracking performance of the FJR utilizing the designed controller. Fig. 9 illustrates the position of the FJR link, while Fig. 10 depicts the angular displacement (α) caused by vibrations in the FJR link.

Figure 9:

Figure 10:

Fig. 9 demonstrates that the link position of the FJR adheres to the desired trajectory satisfactorily, exhibiting a maximum tracking error of approximately 1°. The vibration angle of the FJR link is within the allowed range, with a maximum value of around 0.005°. This observation suggests that the designed controller has successfully followed the trajectory and effectively reduced the vibration of the FJR link.

However, Fig. 2 illustrates the oscillation of the vibration angle of the FJR connection in response to the switching of the square wave. This phenomenon occurs due to the controller’s need to rapidly adjust its output to follow the specified trajectory. The subsequent part of the response exhibits a diminishing oscillatory pattern with an amplitude of less than 0.005o. The oscillation decrease can be attributed to the intrinsic dynamics of the FJR system. Nevertheless, its impact on the tracking position of the tip of the FJR system is not substantial.

Broadly speaking, the results illustrated in the two figures showcase the effectiveness of the applied controller in precise trajectory tracking and vibration reduction for the FJR link. However, room for improvement exists, particularly in reducing the vibration angle during the square-wave transition.

Finally, a comparative study is conducted between the proposed method and other controllers with similar structures as in multiple researches in the literature [17, 20,21,22, 27]. Clearly, it is shown from the zoomed version of (α) and (e₁) in Figs. 6, 7, and 10 that the deflection angle (α) and tracking error (e₁) are lesser and smoother than that obtained in the researches of Dharavath and Ohri [20] and Rsetam et al. [27]. This is because the proposed ISMC-based barrier function can smooth out the control signal by eliminating the harmful chattering that can be noticed in the control input designed by Panchal et al. [17] and Rsetam et al. [22] without losing the precision of the tracking, as shown in Figs. 5 and 9. However, the tracking trajectory of the system encountered in multiple studies in the literature [17, 20,21,22, 27] deviates from the desired reference, and the sliding manifold in the researches of Rsetam et al. [22, 27] suffers from the chattering problem, whereas these problems or limitations are not encountered pursuant to use of the proposed control employed in the present research. More importantly, the proposed controller can impose the end-effector of the link to follow the predefined trajectory better than others, as shown in Figs. 5 and 9.